Introduction

This book is a sequel to [Li1] (hereafter Volume I). Its purpose is to introduce a range of methods, ideas, and techniques that may reasonably be grouped under the broad heading of linear algebra, assuming familiarity with the basic structures of algebra. These methods pervade contemporary mathematics. To address practical needs as comprehensively as possible, the relevant techniques must be forged into purer and more concise forms; categories and functors are an indispensable language for doing so. The book is divided into three main parts: the Inner Part, the Outer Part, and the appendices. It is intended chiefly for advanced undergraduates, graduate students, researchers, teachers, and independent learners who work on or are interested in these subjects. The prerequisites include familiarity with algebraic structures such as groups, rings, modules, and fields, as well as category theory. Volume I covers all of this background, though other good textbooks will serve equally well apart from minor differences in notation. The motivation for writing the book was explained in the introduction to Volume I; the original purpose remains unchanged and need not be repeated here.

What Is Linear Algebra?

In the usage of algebraists, “linear” broadly describes properties connected with structures that have addition and its inverse operation, subtraction. Addition and subtraction in turn give rise to multiplication by integers; an immediate generalization is scalar multiplication by elements of a ring \(R\). The prototypical examples are left and right \(R\)-modules, while a \(\ensuremath{\mathbb{Z}}\)-module is simply an abelian group. For a given \(R\)-module (taken to be a left module unless otherwise stated), one may study its submodules and quotient modules, as well as the kernel \(\operatorname{ker}(f)\), cokernel \(\operatorname{coker}(f)\), and image \(\operatorname{im}(f)\) of a module homomorphism \(f\). All are old friends from elementary algebra; at least the special case in which \(R\) is a field—that is, the case of vector spaces—is widely familiar. Since the twentieth century, mathematical practice has gradually shown that extending the study of \(R\)-modules and their homomorphisms to complexes of \(R\)-modules is not only useful and convenient, but often necessary. Such a complex is a sequence of module homomorphisms satisfying \(d^{n+1}d^n=0\): \[\cdots \longrightarrow X^{n-1} \xrightarrow{d^{n-1}} X^n \xrightarrow{d^n} X^{n+1} \xrightarrow{d^{n+1}} \cdots .\] The condition on the homomorphisms is sometimes written \(d^2=0\), while the full data of the complex are often abbreviated as \((X^n,d^n)_n\), \((X,d)\), or \(X\). Because the notation uses increasing superscripts, this is also called a cochain complex. If instead one uses decreasing subscripts, \(\cdots\to X_n\xrightarrow{d_n}X_{n-1}\to\cdots\), the resulting mathematical object is called a chain complex; the distinction is purely formal. A single \(R\)-module \(M\) may be regarded as the special complex with \(X^0=M\) and all other terms zero. For a complex \(X\), the identity \(d^nd^{n-1}=0\) allows its \(n\)th cohomology to be defined as the quotient module \[\operatorname{H}^n(X):=\operatorname{ker}\left(d^n\right)\big/\operatorname{im}\left(d^{n-1}\right).\] A complex satisfying \(\operatorname{H}^n(X)=0\) for every \(n\) is called an exact sequence or an acyclic complex. For a chain complex, correspondingly, one has homology \(\operatorname{H}_n(X):=\operatorname{ker}(d_n)/\operatorname{im}(d_{n+1})\). The cohomology of a complex (or the homology of a chain complex) often contains important information about the mathematical object under study.

For modules and other mathematical structures equipped with some kind of linear operation—for example, objects of the abelian categories introduced later—the study of complexes and their cohomology is the classical subject matter of homological algebra. This forms a proper part of “linear algebra” in its genuine sense, yet lies at the heart of this book. A brief account of its origins follows.

A Brief History of Homology

Mathematics need not be grounded in history; nevertheless, looking back to understand what lies ahead can only be beneficial. We shall divide the development of homology theory, somewhat artificially, into several stages.

The Formative Period

The first impetus for the development of homological algebra came from topology. In the latter half of the nineteenth century, B. Riemann and E. Betti studied the genus of surfaces and its higher-dimensional generalizations, now called Betti numbers. Around the turn of the century, H. Poincaré developed these ideas into a more rigorous theory. Roughly speaking, Poincaré’s original idea was to decompose a space into polygons, polyhedra, or higher-dimensional analogues glued together; compute Betti numbers and their torsion analogues from matrices encoding the gluing; and prove that these quantities are topological invariants. In a short paper from 1925, E. Noether explained how homology could be defined as an abelian group, with the Betti numbers and their torsion analogues merely numerical invariants derived from it. This already comes close to the modern definition in algebraic topology: for a decomposed space \(E\) and an abelian group \(A\), one defines a chain complex \(C(E,A)\) that records how the pieces are glued; the homology groups of \(E\) with coefficients in \(A\) are then \(\operatorname{H}_n(E;A):=\operatorname{H}_n(C(E,A))\), while the cohomology groups \(\operatorname{H}^n(E;A)\) are defined dually. From then until about 1950, algebraic topology entered an extraordinarily fertile period. From the algebraic point of view, notable developments included:

• the universal coefficient theorem for homology (or cohomology), which says that the case of integer coefficients suffices to determine homology (or cohomology) with coefficients in an arbitrary abelian group \(A\); its explicit description involves the \(\operatorname{Tor}\) (or \(\operatorname{Ext}\)) functor, to be studied in depth later;

• the cup product on cohomology, a multiplicative structure carried by cohomology groups;

• aspherical spaces: connected spaces \(E\) satisfying \(n>1\implies\pi_n(E)=0\), where \(\pi_n(E)\) denotes the \(n\)th homotopy group relative to a chosen base point. W. Hurewicz proved that the homology and cohomology groups of an aspherical space are entirely determined by \(\pi_1(E)\) and can be described algebraically. This marked the beginning of group cohomology.

Another impetus for homological algebra came from within algebra itself. The classical example is Hilbert’s syzygy theorem in invariant theory (1890): in modern language, it states that every module over the polynomial ring \(\Bbbk[X_1,\ldots,X_n]\) has a free resolution of length \(\leq n\). The \(\operatorname{Ext}\) functor likewise arises from algebraic questions. For abelian groups \(A\) and \(B\), a short exact sequence of abelian groups \(0\to A\to E\to B\to0\) is also called a group extension; equivalence classes of such extensions correspond bijectively to the elements of \(\operatorname{Ext}^1(B,A)\). Moreover, if \(B\) is replaced by an arbitrary group \(G\), the classification of group extensions leads naturally to an algebraic definition of group cohomology \(\operatorname{H}^2(G,A)\). Derivations on rings attracted algebraists’ attention early on. In 1942, G. Hochschild defined invariants of a ring \(R\) now called Hochschild homology and cohomology; the degree-one cohomology group classifies derivations modulo inner derivations.¹ These constructions extend to arbitrary \((R,R)\)-bimodules \(M\) and continue to play an important role at the frontiers of modern mathematics. Also worth mentioning is J. Leray’s wartime work on sheaves and spectral sequences, motivated initially by fixed-point problems related to partial differential equations. Sheaves are geometric structures that carry local–global information, while spectral sequences provide powerful tools for computing sheaf cohomology. Influenced by Leray’s theory and Kiyoshi Oka’s work in several complex variables, H. Cartan and others set about rewriting the foundations of algebraic topology after the war; homological algebra thereby took on an entirely new form.

The Cartan–Eilenberg Revolution

The monumental and far-reaching work of H. Cartan and S. Eilenberg [CE56] was long regarded as the canonical reference. It marked a new period in the development of homological algebra. Its main contributions include:

• defining injective and projective modules, and injective and projective resolutions of modules;

• defining right and left derived functors in the framework of module theory, thereby giving general definitions of \(\operatorname{Ext}\) and \(\operatorname{Tor}\), and introducing the injective and projective dimensions of a module—tools that soon proved highly effective in the theory of commutative rings;

• using this machinery to explain earlier constructions, such as group cohomology and Hochschild homology and cohomology;

• systematically establishing the general theory of spectral sequences and their multiplicative structures.

The term “homological algebra” also originates in [CE56]. With the emergence of this grand theory, homological algebra entered its youth.

Abelian Categories

From a geometric—more precisely, sheaf-theoretic—viewpoint, the module-theoretic framework for homological algebra is too restrictive. One seeks categories \(\mathcal A\) more general than the category \(R\ensuremath{\text{-}\mathsf{Mod}}\) of left \(R\)-modules, while retaining some “linear” character. An early attempt appeared in D. Buchsbaum’s 1955 doctoral thesis, also included as an appendix to [CE56]; he abstracted the notion of an exact sequence from module theory. The now standard approach comes from A. Grothendieck’s paper [Gr57]; it led to what are now called abelian categories. An abelian category is an additive category with kernels and cokernels in which every morphism \(f:M\to N\) is strict. Roughly speaking, strictness corresponds to the familiar module-theoretic isomorphism theorem \(M/\operatorname{ker}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}(f)\). In the same work, Grothendieck also:

• used long exact sequences to define \(\delta\)-functors and universal \(\delta\)-functors, characterized the universal ones, and showed that derived functors are special cases of universal \(\delta\)-functors;

• gave the Grothendieck spectral sequence for deriving a composite of functors;

• defined a special class of abelian categories, now called Grothendieck categories, and established the existence of injective resolutions in them.

These theories made it possible to study sheaf cohomology on more general spaces and were crucial to the development of geometry.

Derived Categories

In his 1963 doctoral thesis, J.-L. Verdier defined the derived category \(\ensuremath{\mathsf{D}}(\mathcal A)\) of an abelian category \(\mathcal A\). The motivation came from difficulties Grothendieck encountered while studying duality in algebraic geometry. Those obstacles prompted the search for a framework in which one works with complexes themselves rather than merely with their cohomology, formally inverting the quasi-isomorphisms in the category of complexes \(\ensuremath{\mathsf{C}}(\mathcal A)\). A quasi-isomorphism is a morphism of complexes \(f:X\to Y\) inducing isomorphisms \(\operatorname{H}^n(f):\operatorname{H}^n(X)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{H}^n(Y)\) on cohomology. The operation of formally inverting these morphisms is called Gabriel–Zisman localization, a categorical generalization of ring localization. The derived category \(\ensuremath{\mathsf{D}}(\mathcal A)\) contains more information than cohomology alone, and its formalism is correspondingly different: sequences of morphisms in \(\ensuremath{\mathsf{D}}(\mathcal A)\) called distinguished triangles, \[X \xrightarrow{f} Y \xrightarrow{g} Z \xrightarrow{h} X[1]\] play the role of short exact sequences in \(\mathcal A\) or in \(\ensuremath{\mathsf{C}}(\mathcal A)\). Here \(X[1]\) is the shift of the complex \(X\): \[X[1]^n=X^{n+1},\quad d_{X[1]}^n=-d_X^n.\] Verdier further abstracted these constructions into structures called triangulated categories. From the topological viewpoint, the triangulated structure of a derived category is equally natural: distinguished triangles are analogous to cofiber sequences in homotopy theory. This relates to the homotopical algebra discussed later. Also motivated by homotopy theory, D. Puppe independently discovered a similar structure in 1962, although his conditions did not include Verdier’s octahedral axiom. Although derived categories are not ideal for every application, their language remained the preferred one among geometers and algebraists into the early twenty-first century. One prominent application is the theory of \(\mathscr D\)-modules, developed chiefly by Masaki Kashiwara, Z. Mebkhout, and others; some of its basic operations admit satisfactory definitions only at the level of derived categories. The derived category of a ring \(R\), \(\ensuremath{\mathsf{D}}(R):=\ensuremath{\mathsf{D}}(R\ensuremath{\text{-}\mathsf{Mod}})\), may also be regarded as an invariant of \(R\). If two rings have equivalent derived categories as triangulated categories, they are called derived equivalent. Questions and methods surrounding derived equivalence form a major current in the representation theory of algebras, pioneered by D. Happel, J. Rickard, and others.

Homotopical Algebra

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1. Translator’s note: the source says that the degree-one term “classifies all derivations.” This has been corrected here: \(\operatorname{H}^1_{\mathrm{Hoch}}(R,R)\) classifies derivations modulo inner derivations; see correction O014-C001.↩︎

The Purpose of This Book

Organizational Aims

The preceding introduction shows that what is called linear algebra divides roughly into two main branches: the theory of abelian categories and the theory of complexes built upon them. This book centers on these two branches and seeks to provide the ideas and techniques needed to understand contemporary mathematics. Its contents include both the algebraic methods used in applications and the theoretical foundations those methods require. Since the methods primarily serve research in pure mathematics and are meant to apply as broadly as possible, the book may also be characterized as “pure applied mathematics.” The book also attempts to serve both as a textbook and as a reference, which necessarily calls for attention to detail and systematic development. These goals pull against one another, yet the book must state its purpose, develop its themes, and bring them home within a finite space. This creates an enormous organizational challenge, but one that must be accepted. As to content, derived categories, derived functors, and spectral sequences are essential knowledge for contemporary mathematicians. Yet they pose real difficulties for beginners, chiefly at the level of ideas and definitions. The Inner Part is organized around this objective and lays the foundation for the Outer Part. To keep the exposition flowing and save space, we shall not retrace history step by step, but will more often follow the logical order made visible by hindsight. Constrained by these aims, what critics of art and literature call an “aura” inevitably recedes from the prose, replaced at times by the heaviness of systematic structure and rigorous proof. The aura opposed to that heaviness is mathematics’ first knock upon the human mind, and perhaps its last. In the author’s view, however, a light, agile, direct style belongs either to a theory’s formative period or relies upon a complete supporting literature. Such support must be embedded in the native linguistic culture and shared by every reader. The author has chosen to carry the burden, hoping that the appearance of this book will leave future authors free to travel more lightly. During its preparation, the book drew extensively on related works, among them [Wei94, ML98, KS06, Rie16, Yek20]. The Stacks Project [stacks] and the nLab website also proved invaluable.

Approach and Trade-offs

The book must accommodate many fields and styles while finding a balance between classical and modern approaches, so that the material remains reasonably accessible to beginners. Compromise is therefore the author’s guiding principle. Few readers, however, open a book in search of compromise. Compromise brings detours, so the path of this book is a geodesic for no reader; that is unavoidable. For the same reason, the selection of material cannot satisfy everyone. Geometrically inclined readers will not find sheaves or \(t\)-structures, while readers interested in the representation theory of algebras will not find tilting theory. These topics are omitted because the corresponding theories cannot be adequately motivated without first entering those subjects more deeply. Category theorists may regard the book’s approach as no more than semiclassical. Even so, readers may notice that the main text and exercises leave entry points for more advanced viewpoints, such as dg-categories, monad theory, and simplicial objects, though each is touched on only briefly. These topics can be viewed as a warm-up for infinity-category theory and point toward a new world deeper than linear algebra, called advanced or higher algebra, which lies beyond the scope of this book. One further choice deserves explanation: the book treats general abelian categories and develops their basic properties directly from the axioms, unlike some homological-algebra texts that consider only modules. This generality is demanded both by theoretical completeness and by practical necessity: even if modules were our sole starting point, constructions such as functor categories and Serre quotients would still require it. Because diagram chases in module theory do not extend directly to the general setting, some foundational results about abelian categories require rather indirect proofs. To help with these details, the book expects readers to have some elementary experience with complexes of modules and their cohomology, such as the material of [Li1], though this is not logically necessary. The celebrated Freyd–Mitchell embedding theorem states that every small abelian category admits an exact, fully faithful embedding into \(\ensuremath{\mathsf{Mod}\text{-}}R\) for some ring \(R\). Granting this theorem, many basic facts about abelian categories can be checked in the concrete module category \(\ensuremath{\mathsf{Mod}\text{-}}R\). A proof of the theorem is given in Appendix

§B.6. Because the proof is itself nontrivial and depends on basic properties of abelian categories, the theorem does not essentially simplify their theory; its role is chiefly heuristic.

Guide to Using the Book

How the Book May Be Used

To serve as a reference work, the book needs a reasonably systematic organization. Its chapters therefore follow a primarily logical order. Dependencies between earlier and later material are especially pronounced in the Inner Part, while the few necessary topics that diverge from the main line or are more technical have been assigned to appendices. Independent readers should decide what to read according to their own interests and needs. Most of these choices arise in the Outer Part, whereas the Inner Part supplies the common foundational framework, although even there some sections are optional. More abstract or technical details may be omitted on an initial reading, while material involving the appendices is best skipped at first or merely skimmed. Detailed advice on these choices appears at the beginning of each chapter and appendix, especially in the reading guides. As noted above in the discussion of the book’s aims, we have unavoidably wrapped mathematical ideas in layer upon layer of prose. Independent readers must make a determined effort to discern the mathematics itself within those layers, like catching lightning amid roiling clouds. For readers using the book as a reference, the author has tried to organize the exposition modularly so that its parts can be read separately. This is reflected not only in the division into chapters, but also in the following two features.

• As far as possible, the book uses the standard notation of contemporary literature, supplemented by reviews and explanations. The Conventions at the end of the introduction and the index at the end of the book provide two further aids to understanding the notation.

• Statements and proofs contain copious cross-references, allowing readers who already know something of the subject to understand the discussion quickly and build an efficient conceptual map.

In particular, the author hopes that the book will help readers preparing to teach or participating in seminars to assemble the knowledge they need quickly and then recast it in their own language. As for classroom use, one must frankly admit that teaching schedules are not governed by individual wishes. While writing the book, the author had almost no opportunity to try out this material in the classroom, so its effectiveness there remains unknown. Judging from past experience, even with details omitted in lectures, the book clearly contains enough material for more than one academic year; in a streamlined presentation, the Inner Part might perhaps be compressed into a single semester. An instructor using the book as the primary textbook will first have to decide how to divide the material. The author hopes that the modular design just mentioned will make this task easier. The lack of classroom testing has two further consequences. First, the book must still contain many undiscovered errors. Second, it lacks the finish of a classic textbook. This is due both to practical circumstances and to the author’s own limitations; the author can only ask the reader’s indulgence.

On the Exercises

Every chapter and appendix ends with exercises arranged in the order of the corresponding material in the main text and sometimes accompanied by hints. Their chief purposes are to supplement the text with further properties, provide examples, and broaden the reader’s perspective. Some exercises help readers become familiar with techniques and generally make few technical demands. A few ask readers to fill in details omitted from the text; these details are either trivial, easy, or, more often, both. In short, the author hopes that beginners will attempt as many exercises as possible, but they need not force themselves or feel obliged to complete them all.

Categories and Their Size

Most of this book is expressed in categorical language because practice has shown category theory to be an efficient means of understanding and handling algebraic problems. Nevertheless, linear algebra—or what is called homological algebra—is no more a branch of category theory than probability theory is a branch of measure theory. Unlike some textbooks, both this book and Volume I require that, for a category, the collection of all objects and the collection of all morphisms each be a set; it is not enough for the objects merely to form a “class.” The price of excluding proper classes from the definitions is that, strictly speaking, one must introduce Grothendieck universes to measure the size of sets and assume that all sets belong to some universe \(\mathcal{U}\); see [Li1]. When concrete problems are considered case by case, the axiom that there are sufficiently many universes—equivalently, sufficiently many strongly inaccessible cardinals—can often be weakened. Since this issue is not central to the book, we shall not pursue it further.

Overview of the Structure

Readers are advised to begin with the Conventions at the end of the introduction, in order to familiarize themselves with the notation and conventions used throughout the book. Beyond that, beginners are discouraged from reading strictly in sequence unless they are already highly receptive to abstract methods. Each chapter opens with a detailed summary and reading guide. First comes the Inner Part, which lays the foundations.

• Chapter 1: Supplements to Category Theory.Reviews and rounds out the requisite category theory, emphasizing strict morphisms, kernels and cokernels in additive categories, Kan extensions, Gabriel–Zisman localization, and related topics.

• Chapter 2: Abelian Categories.Develops the basic theory of abelian categories, supplemented by material on Grothendieck categories and other topics.

• Chapter 3: Complexes.Discusses basic notions such as complexes, double complexes, homotopies, and mapping cones in additive categories, as well as cohomology in abelian categories. The second half treats derived functors from the classical viewpoint, including special cases such as \(\operatorname{Ext}\), \(\operatorname{Tor}\), and \(\lim^1\). To handle unbounded complexes, it concludes with K-injective and K-projective resolutions.

• Chapter 4: Triangulated and Derived Categories.Introduces the general notions of triangulated categories and triangulated functors, then concentrates on derived categories of abelian categories and derived functors between derived categories. The classical \(\operatorname{Ext}\) and \(\operatorname{Tor}\) are upgraded accordingly to \(\operatorname{RHom}\) and \(\ensuremath{\underset{}{\overset{\mathrm{L}}{\otimes}}}\).

• Chapter 5: Spectral Sequences.Begins with the general definition of spectral sequences and exact couples, then specializes progressively to the spectral sequences of filtered complexes and double complexes and to their applications in algebra. Exact couples are not absolutely necessary for the spectral sequence of a filtered complex.

Next comes the Outer Part, devoted to applications or further developments.

• Chapter 6: Group Homology and Cohomology.Studies group homology and cohomology, together with a variety of related computations and examples. It also treats Tate cohomology, the cohomology of profinite groups, and nonabelian cohomology, each of which plays an important role in applications.

• Chapter 7: Monad Theory.Starts from the general notion of an “algebra” in a monoidal category and first discusses differential graded structures related to complexes. After treating examples that include coalgebras, bialgebras, and Hopf algebras, it turns to Beck’s monadicity theorem. The second half considers module theory, introduces Morita theory, and applies monadicity to flat descent for modules and to Galois descent.

• Chapter 8: Simplicial Methods.Covers the general theory of simplicial objects and simplicial sets, the Dold–Kan correspondence, and the bar construction formulated in the language of monads. The final sections concern the generalized Eilenberg–Zilber theorem and a topological interpretation of mapping cones.

• Chapter 9: Duality.Begins with duality in monoidal categories and, after basic examples, studies reconstruction theorems and Tannakian categories. This chapter is not directly concerned with complexes.

The original plan called for the Outer Part to include the foundations of representation theory, which seemed likely to play a natural role in the overall structure. Late in the process, the idea was abandoned because the book was already too long and several mature textbooks were available. The appendices are as follows.

• Appendix A: Further Material on Abelian Categories.Collects material directly or indirectly related to abelian categories, either cited in the main text or developed as a side topic.

• Appendix B: An Introduction to Ind- and Pro-objects.Starts from the theory of profinite groups, naturally introduces ind- and pro-objects, and studies the Ind-construction for categories and functors. As an application, it then proves the Freyd–Mitchell embedding theorem for abelian categories.

Finally, the author’s personal website provides the latest supplementary information, including errata, for this book and the author’s other works. Readers are invited to follow the updates there and to write to the author if they find any errors in the book.

Acknowledgments

During the preparation of this book, the author benefited from corrections and suggestions by teachers, colleagues, and friends. The process lasted so long that omissions and errors of memory are inevitable; the following list is incomplete: Huan Zhen, Huang Chenxin, Li Jinghui, Lei Jiale, Li Changyuan, Sun Yuze, Tang Yiming, Xue Hanyu, Yang Enlin, Zhang Haofeng, and Zhao Zelong (in Hanyu Pinyin order). A small portion of the book was used in courses at the University of Chinese Academy of Sciences and Huazhong University of Science and Technology. The author also thanks everyone who took part in those courses. Like all the author’s works to date, this book was written entirely in an open-source software environment, including the operating system and the fonts used in preparing it for press. The author salutes the many selfless developers who put into practice the universal values of sharing and altruism. While this book was being written, the author’s research was supported by the Excellent Young Scientists Fund of the National Natural Science Foundation of China under project 11922101, with Peking University as the host institution. This support is gratefully acknowledged. The author also thanks Zhao Tianfu, an editor at Higher Education Press, for his patience and professionalism. The writing of this book ought to have followed seamlessly from Volume I, but various obstacles delayed its completion until the end of 2022. Despite the delay, preparing the book consumed even more of the author’s mental energy than Volume I. This was especially true of the repeated checking and revision in the final stages, which felt like walking through an endless dark tunnel. Much of this late work was done in Beijing subway cars. Working on the train was hardly comfortable and required making every minute and second count, quite literally. Yet amid that immense, anonymous current, day after day, the author seemed to sense a kind of tacit mutual understanding rarely found in working life, and to draw a glimmer of consolation from it. Was it so, or was it not? The explanation may be fanciful, but the feeling was real. The prelude now draws to a close: homage to the boundless distance, and to the endless multitudes.

Conventions

The notational conventions in this book are the same as in Volume I. They are summarized here, at the risk of some length.

Basic Conventions

The book uses standard logical symbols such as \(\forall\), \(\exists\), and \(\implies\). The symbol \(\exists !\) means “there exists exactly one,” and \(\iff\) means “if and only if.” The end of a proof is marked by \(\Box\). The notation \(A:=B\) means “\(A\) is defined to be \(B\).” If the definition of a mathematical object does not depend on choices of auxiliary data, it is called well-defined.

Sets and Order Structures

The book uses Zermelo–Fraenkel axiomatic set theory and accepts the axiom of choice. The empty set is denoted by \(\emptyset\). The notation \(A\subset B\) means that \(A\) is a subset of \(B\), with equality allowed; strict inclusion is written \(A\subsetneq B\). Set difference is \(A\smallsetminus B:=\{a\in A:a\notin B\}\), and disjoint union is \(A\sqcup B\). For a map \(f:A\to B\), the notation \(a\mapsto b\) means that \(a\in A\) is mapped by \(f\) to \(b\in B\), and \(f|_{A'}\) is the restriction of \(f\) to a subset \(A'\subset A\). The identity map on \(A\) is denoted by \(\ensuremath{\mathrm{id}}=\ensuremath{\mathrm{id}}_A\). The number of elements of \(A\), also called its cardinality or cardinal, is denoted by \(|A|\). Following [Li1], a partially ordered set

is a set \(P\) equipped with a reflexive, transitive, and antisymmetric binary relation \(\leq\). Omitting antisymmetry gives a preordered set. The notation \(x<y\) means \(x\lneq y\). A partially ordered set \((P,\leq)\) is often abbreviated to \(P\).

• If \(m\in P\) satisfies \(\forall x\in P,\;x\geq m\iff x=m\), then \(m\) is a maximal element of \(P\). Replacing \(\geq\) by \(\leq\) defines a minimal element. In general these need not be unique.

• Let \(S\) be a subset of \(P\). If \(b\in P\) satisfies \(\forall x\in S,\;x\leq b\), then \(b\) is an upper bound of \(S\) in \(P\). Lower bounds are defined similarly.

• If \(b\) is an upper bound of \(S\) and every other upper bound \(b'\) satisfies \(b'\geq b\), then \(b\) is the supremum of \(S\), denoted \(\sup S\). A supremum, if it exists, is unique. The infimum \(\inf S\) is defined similarly.

A map \(f:P\to Q\) of partially ordered sets is order-preserving if \(a\leq b\implies f(a)\leq f(b)\). If \(f\) is bijective and both \(f\) and \(f^{-1}\) are order-preserving, it is an isomorphism of partially ordered sets, or an order-preserving bijection. If every two elements of a partially ordered set \(P\) are comparable, then \(P\) is totally ordered. A nonempty totally ordered set in which every nonempty subset has a least element is well-ordered . Certain special well-ordered sets are called ordinals, and any two ordinals are comparable. Briefly:

• for every ordinal \(\alpha\), one has \(\alpha=\{\beta:\text{ordinal},\;\beta<\alpha\}\);

• if an ordinal \(\kappa\), as a set, is not equinumerous with any ordinal smaller than \(\kappa\), then it is called a cardinal;

• the cardinality \(|S|\) of a set \(S\) is defined as the least ordinal in its equinumerosity class. This uses the fact that every set can be well-ordered (which requires the axiom of choice) and that every well-ordered set is isomorphic to a unique ordinal;

• in particular, \(|\alpha|\leq\alpha\) for every ordinal \(\alpha\).

This is von Neumann’s viewpoint; see [Li1] or [Je03]. It is slightly more circuitous than the usual definition of a cardinal as an equinumerosity class, but has its conveniences. Finite ordinals may be identified with nonnegative integers: for each \(n\in\ensuremath{\mathbb{Z}}_{\geq0}\) there is a corresponding totally ordered set \(\mathbf n\), recursively defined as a set by \(\mathbf0:=\emptyset\) and \(\mathbf{n+1}:=\{\mathbf0,\ldots,\mathbf n\}\), with the evident order. The least infinite ordinal \(\omega\) is the well-ordered set \(\mathbf0\leq\mathbf1\leq\cdots\), and is also the countable cardinal \(\aleph_0\).

We choose a Grothendieck universe

\(\mathcal U\) to distinguish sizes of sets. Elements of \(\mathcal U\) are called \(\mathcal U\)-sets; sets equinumerous with a \(\mathcal U\)-set are called \(\mathcal U\)-small, or simply small sets. If a cardinal \(\mu\), regarded as a set, is \(\mathcal U\)-small, it is a small cardinal. We assume that every set \(X\) belongs to some \(\mathcal U\); see [Li1] and the accompanying discussion.

Categories

Our conventions for categories agree with [Li1]: the objects and morphisms of a category \(\mathcal C\) each form a set, denoted by \(\operatorname{Ob}(\mathcal C)\) and \(\operatorname{Mor}(\mathcal C)\). The set of morphisms between two objects is denoted by \(\operatorname{Hom}_{\mathcal C}(\cdot,\cdot)\), or simply \(\operatorname{Hom}(\cdot,\cdot)\). The identity morphism of \(X\) is \(\ensuremath{\mathrm{id}}_X\). Monomorphisms are marked by \(\hookrightarrow\) and epimorphisms by \(\twoheadrightarrow\); a monomorphism is also called an embedding.

• For any morphism \(f:X\to Y\) in a category and any object \(T\), \(f\) induces pullback \(f^*\) and pushforward \(f_*\) operations on \(\operatorname{Hom}\) sets: \[\begin{align*} f^*: \operatorname{Hom}(Y, T) & \to \operatorname{Hom}(X, T), & f_*: \operatorname{Hom}(T, X) & \to \operatorname{Hom}(T, Y) \\ \beta & \mapsto \beta f & \alpha & \mapsto f\alpha . \end{align*}\]

• Subcategories are written using subset notation, \(\mathcal C'\subset\mathcal C\). If \(\operatorname{Hom}_{\mathcal C'}(X,Y)=\operatorname{Hom}_{\mathcal C}(X,Y)\), then \(\mathcal C'\) is a full subcategory.

• The product \(\mathcal C_1\times\mathcal C_2\) of two categories has object set \(\operatorname{Ob}(\mathcal C_1)\times\operatorname{Ob}(\mathcal C_2)\). A morphism from \((X_1,X_2)\) to \((Y_1,Y_2)\) is an element of \(\operatorname{Hom}_{\mathcal C_1}(X_1,Y_1)\times\operatorname{Hom}_{\mathcal C_2}(X_2,Y_2)\), and composition is componentwise. This immediately defines products \(\prod_i\mathcal C_i\) of arbitrary families of categories. Likewise, \(\mathcal C^n:=\mathcal C\times\cdots\times\mathcal C\) (\(n\) factors), or more generally \(\mathcal C^I\) for a set \(I\). See [Li1].

For example, every preordered set \((P,\leq)\) determines a category \(\mathcal P\) with object set \(P\) and \[\forall a,b \in P, \quad \operatorname{Hom}_{\mathcal{P}}(a, b) = \begin{cases} \text{the singleton} \; \{ \star \}, & a \leq b \\ \emptyset, & \text{otherwise.} \end{cases}\] In particular, \(\mathbf0\) may be identified with the empty category, having no objects and no morphisms, while \(\mathbf1\) is identified with the category having exactly one object \(\mathbf0\) and one morphism \(\ensuremath{\mathrm{id}}_{\mathbf0}\). A category satisfying \(\operatorname{Mor}(\mathcal C)=\{\ensuremath{\mathrm{id}}_X:X\in\operatorname{Ob}(\mathcal C)\}\) is called discrete. Discrete categories correspond bijectively to sets under \(\mathcal C\mapsto\operatorname{Ob}(\mathcal C)\).

Terminology: Small and Large Categories

Fix a Grothendieck universe \(\mathcal U\), and let \(\mathcal C\) be a category. If \(\mathrm{Mor}(\mathcal C)\) is \(\mathcal U\)-small, then \(\mathcal C\) is a \(\mathcal U\)-small category. If only \(\operatorname{Hom}_{\mathcal C}(X,Y)\) is required to be \(\mathcal U\)-small for every \(X,Y\in\operatorname{Ob}(\mathcal C)\), then \(\mathcal C\) is a \(\mathcal U\)-category. ¹ For example, all \(\mathcal U\)-sets and maps between them form the \(\mathcal U\)-category \(\ensuremath{\mathsf{Set}}\), which is not \(\mathcal U\)-small. Without a choice of \(\mathcal U\), on the other hand, all sets form a proper class and not a category in the sense of this book. Similarly, \(\ensuremath{\mathsf{Grp}}\) (or \(\ensuremath{\mathsf{Ab}}\)) denotes the category of all groups (or abelian groups) realized on \(\mathcal U\)-sets; these are \(\mathcal U\)-categories.

To see that size is a genuine issue, observe that the \(\operatorname{Hom}\) functor \(\operatorname{Hom}_{\mathcal C}:\mathcal C^{\operatorname{op}}\times\mathcal C\to\ensuremath{\mathsf{Set}}\) can be defined only when \(\mathcal C\) is a \(\mathcal U\)-category. Unless otherwise stated, categories in this book are \(\mathcal U\)-categories, and \(\mathcal U\)-small categories are simply called small. A category not necessarily a \(\mathcal U\)-category is called large.

Large categories are hard to avoid in many constructions, especially in theories involving functor categories or localization. Whenever a large category may occur and has substantive consequences, this will be stated explicitly.

Functors and Functor Categories

The identity functor of \(\mathcal C\) is denoted by \(\ensuremath{\mathrm{id}}_{\mathcal C}\). A functor \(F:\mathcal C\to\mathcal D\) sends a morphism \(f:X\to Y\) in \(\mathcal C\) to a morphism \(Ff:FX\to FY\) in \(\mathcal D\). If \(\operatorname{Hom}_{\mathcal C}(X,Y)\to\operatorname{Hom}_{\mathcal D}(FX,FY)\) is injective (respectively bijective) for every \(X,Y\), then \(F\) is faithful (respectively fully faithful). If every object of \(\mathcal D\) is isomorphic to some \(FX\), then \(F\) is essentially surjective.

For categories associated with partially ordered sets, an order-preserving map \(P\to Q\) is precisely a functor \(\mathcal P\to\mathcal Q\).

All functors from \(\mathcal C\) to \(\mathcal D\) form the functor category \(\mathcal D^{\mathcal C}\), also denoted \(\mathrm{Fct}(\mathcal C,\mathcal D)\). Unless \(\mathcal C\) is small, \(\mathcal D^{\mathcal C}\) is often a “large category.” As special cases, for \(n\in\ensuremath{\mathbb{Z}}_{\geq0}\) and the corresponding totally ordered set \(\mathbf n\), the categories \(\mathcal C^{\mathbf n}\) are described as follows.

• by definition, \(\mathcal C^{\mathbf0}:=\mathbf1\);

• there is a unique functor \(\mathcal C\to\mathbf1\), hence \(\mathbf1^{\mathcal C}=\mathbf1\);

• specifying a functor \(\mathbf1\to\mathcal C\) is equivalent to specifying an object of \(\mathcal C\), hence \(\mathcal C^{\mathbf1}\simeq\mathcal C\);

• the objects of \(\mathcal C^{\mathbf2}\) are morphisms in \(\mathcal C\), and its morphisms are commutative squares between them.

Thus \(\mathbf0\) may be called the initial category, \(\mathbf1=[\bullet]\) the terminal category, and \(\mathbf2=[\bullet\to\bullet]\) may be pictured as a walking arrow.

The opposite category \(\mathcal C^{\operatorname{op}}\) has the same objects as \(\mathcal C\). Every morphism \(f:X\to Y\) may be regarded as a morphism \(Y\to X\) in \(\mathcal C^{\operatorname{op}}\) and is then written \(f^{\operatorname{op}}\). Moreover, \(\mathrm{Fct}(\mathcal C,\mathcal D)^{\operatorname{op}}\) is naturally identified with \(\mathrm{Fct}(\mathcal C^{\operatorname{op}},\mathcal D^{\operatorname{op}})\). Passing from \(\mathcal C\) to \(\mathcal C^{\operatorname{op}}\) reverses arrows; this mechanism is called duality in category theory. If functors \(F:\mathcal C\leftrightarrows\mathcal D:G\) satisfy \(GF\simeq\ensuremath{\mathrm{id}}_{\mathcal C}\) and \(FG\simeq\ensuremath{\mathrm{id}}_{\mathcal D}\), they are mutually quasi-inverse. A functor having a quasi-inverse is an equivalence of categories, a fundamental notion in category theory. If the strict equalities \(GF=\ensuremath{\mathrm{id}}_{\mathcal C}\) and \(FG=\ensuremath{\mathrm{id}}_{\mathcal D}\) hold, then they are mutually inverse isomorphisms of categories.

Algebraic Structures

The identity element of a group is denoted by \(1\) in multiplicative notation and by \(0\) for an abelian group in additive notation. The opposite group, with reversed multiplication, is \(G^{\operatorname{op}}\). If \(H\) is a subgroup of \(G\), its index is \((G:H)\), also equal to the number of cosets \(|G/H|\) or \(|H\backslash G|\).

Unless otherwise stated, rings are associative and unital, as are algebras over commutative rings. The multiplicative identity of \(R\) is \(1\) or \(1_R\), and its invertible elements form the multiplicative group \(R^\times\). The characteristic of a field or integral domain \(R\) is \(\mathrm{char}(R)\). The opposite ring \(R^{\operatorname{op}}\) has multiplication in the reverse order.

Let \(r\) be a nonzero element of a commutative ring \(R\). If the additive-group homomorphism \(R\xrightarrow{\text{multiplication by}\;r}R\) has a nonzero kernel, then \(r\) is a zero divisor. The ideal generated by \(x,y,\ldots\) in a commutative ring is denoted by \((x,y,\ldots)\). The polynomial algebra over a commutative ring \(R\) in variables \(X_1,X_2,\ldots\) is \(R[X_1,X_2,\ldots]\), and the formal power-series ring is \(R\llbracket X_1,X_2,\ldots\rrbracket\). If \(M\) is a monoid, its monoid algebra over \(R\) is \(R[M]\). By convention, \(\ensuremath{\mathbb{Z}}\subset\ensuremath{\mathbb{Q}}\subset\ensuremath{\mathbb{R}}\subset\ensuremath{\mathbb{C}}\) denote, respectively, the ring of integers, the field of rational numbers, the field of real numbers, and the field of complex numbers. If \(q\) is a prime power, \(\ensuremath{\mathbb{F}}_q\) is the finite field with \(q\) elements. If \(F\hookrightarrow E\) is an embedding of fields, the corresponding field extension is denoted by \(E|F\) and its degree by \([E:F]\). If \(E|F\) is Galois, its Galois group is \(\operatorname{Gal}(E|F)\).

The common categories associated with some basic algebraic structures are denoted as follows.

If \(\Bbbk\) is fixed and \(R,S\) are \(\Bbbk\)-algebras, then an \((R,S)\)-bimodule is understood by default to arise from a left \(R\ensuremath{\underset{\Bbbk}{\otimes}}S^{\operatorname{op}}\)-module structure. In other words, the left and right actions of \(\Bbbk\) are required to agree. The group of homomorphisms between two left \(R\)-modules or two right \(R\)-modules is denoted by \(\operatorname{Hom}_R(X,Y)\). For \((R,S)\)-bimodules, the analogous notation is \(\operatorname{Hom}_{(R,S)}(X,Y)\). For a commutative ring \(R\) and a multiplicative subset \(U\), the corresponding localization is \(R[U^{-1}]\). As with the category of sets \(\ensuremath{\mathsf{Set}}\), groups, rings, modules, and the other structures here are understood to be realized on \(\mathcal U\)-sets for the chosen Grothendieck universe \(\mathcal U\). Every category in the table therefore has a forgetful functor to \(\ensuremath{\mathsf{Set}}\). Other examples used in this book, such as the category \(\ensuremath{\mathsf{Top}}\) of topological spaces and the category \(\ensuremath{\mathsf{CHaus}}\) of compact Hausdorff spaces, obey the same convention: their spaces are realized on \(\mathcal U\)-sets.

Adjunctions

Limits

Consider a functor \(\alpha:I\to\mathcal C\).

• The inductive limit of \(\alpha\), if it exists, is denoted by \(\varinjlim\alpha\) or \(\varinjlim_{i\in\operatorname{Ob}(I)}\alpha(i)\). It is an object of \(\mathcal C\) equipped with morphisms \(\alpha(i)\to\varinjlim\alpha\) and determined by a universal property.

• Similarly, the projective limit of \(\alpha\), if it exists, is denoted by \(\varprojlim\alpha\) or \(\varprojlim_{i\in\operatorname{Ob}(I)}\alpha(i)\). It is equipped with morphisms \(\varprojlim\alpha\to\alpha(i)\) and determined by a universal property.

The two kinds of limits are dual: the \(\varinjlim\) of \(\alpha:I\to\mathcal C\) is the \(\varprojlim\) of \(\alpha^{\operatorname{op}}:I^{\operatorname{op}}\to\mathcal C^{\operatorname{op}}\). For this reason, functors involved in \(\varprojlim\) are often written in the form \(I^{\operatorname{op}}\to\mathcal C\). In much of the literature, \(\varinjlim\) is called a colimit and denoted by \(\mathrm{colim}\), while \(\varprojlim\) is called a limit and denoted by \(\lim\). Unless otherwise stated, “limit” in this book includes both \(\varinjlim\) and \(\varprojlim\).

If \(I\) is small, the corresponding \(\varinjlim\) or \(\varprojlim\) is a small limit. Following the standard terminology of [Li1], a category \(\mathcal C\) having all small \(\varprojlim\) (respectively all small \(\varinjlim\)) is complete (respectively cocomplete). This notion depends on the choice of \(\mathcal U\).

To fix notation, choose a category \(\mathcal C\) and review several common special cases of limits. Detailed definitions may be found in [Li1] or other textbooks. In the universal properties below, \(T\) is an arbitrary object of \(\mathcal C\) and \(I\) is an arbitrary set.

Translator’s note. In the coproduct row, the source gives the target of the canonical injection as \(\prod_{j\in I}X_j\). It has been corrected to \(\coprod_{j\in I}X_j\), in agreement with the preceding object column and the following universal property; see correction O014-C003.

Finite products (respectively coproducts) are also written \(X_1\times X_2\times\cdots\) (respectively \(X_1\sqcup X_2\sqcup\cdots\)). If \(\mathcal C\) is additive, the coproduct \(\coprod_i\) is often written using the direct-sum symbol \(\bigoplus_i\) from module theory.

Equalizers and coequalizers, and products and coproducts, form dual pairs. These universal-property characterizations may also be written as commutative diagrams as in [Li1], or interpreted through the Yoneda embedding reviewed in §A.1. Suppose that \(\coprod_{i\in I}X_i\) and \(\prod_{j\in J}Y_j\) exist. Their universal properties give a bijection \[\operatorname{Hom}\left( \coprod_{i \in I} X_i, \prod_{j \in J} Y_j \right) \xrightarrow{1:1} \prod_{\substack{i \in I \\ j \in J}} \operatorname{Hom}(X_i, Y_j), \quad \psi \mapsto (p_j \psi \iota_i)_{i,j} .\]

• Terminal object.A product corresponding to \(I=\emptyset\), if it exists, is a terminal object of \(\mathcal C\), temporarily denoted by \(Z\). By convention, \(\prod_{i\in\emptyset}:=\{\emptyset\}\), so the universal property of \(Z\) says that \(\operatorname{Hom}(T,Z)\) is a singleton for every \(T\in\operatorname{Ob}(\mathcal C)\).

• Initial object.A coproduct corresponding to \(I=\emptyset\), if it exists, is an initial object of \(\mathcal C\), temporarily denoted by \(S\). Its universal property says that \(\operatorname{Hom}(S,T)\) is a singleton for every \(T\in\operatorname{Ob}(\mathcal C)\).

• Zero object and zero morphism.If an object \(0\) of \(\mathcal C\) is both initial and terminal, it is called a zero object; morphisms to and from it exist and are unique. For \(X,Y\in\operatorname{Ob}(\mathcal C)\), the composite \(X\to0\to Y\) gives a canonical element of \(\operatorname{Hom}(X,Y)\) called the zero morphism.

Because an isomorphism from any object \(X\) to a zero object, if it exists, is unique, one customarily says that \(X\) is a zero object or writes \(X=0\), rather than saying that \(X\) is isomorphic to a zero object. We next review two important, mutually dual kinds of limit.

These are called a pullback diagram and a pushout diagram, respectively. The morphism \(X\ensuremath{\underset{Z}{\times}}Y\to Y\) is called the pullback of \(X\to Z\), and \(Y\to X\ensuremath{\underset{Z}{\sqcup}}Y\) the pushout of \(Z\to X\); the roles of \(X\) and \(Y\) are symmetric. More generally, commutative diagrams isomorphic to these are also called pullback and pushout diagrams; see [Li1].

In this book, the center of the square in a pullback (respectively pushout) diagram is marked by \(\Box\) (respectively \(\boxplus\)).

Translation

In the Chinese source edition, the conventions for translation agree with [Li1]; Chinese terminology is kept compatible with [ZG] unless that work’s translation is clearly inappropriate or erroneous. In the source index, Chinese terms are accompanied by English equivalents. This helps readers who need to read or write articles in English, provides a reference for readers whose native language is not Chinese, and, finally, reflects the historical fact that knowledge of foreign languages remains necessary for understanding the meaning of many mathematical symbols.

Translator’s note. This English edition indexes English headwords and records the Chinese–English terminological choices in a separate production glossary.

---

1. In much of the literature, the objects of a category are understood to form a class, while the \(\operatorname{Hom}\) between any two objects is a set. Thus their “classes” correspond to this book’s sets, their “sets” to this book’s \(\mathcal U\)-sets, and their “categories” essentially to this book’s \(\mathcal U\)-categories.↩︎

Supplements to Category Theory

As its title suggests, this chapter introduces several tools from category theory that will be needed in later chapters, both as a review and as an opportunity to see familiar ideas afresh. Some of this material appears in [Li1] or similar textbooks; it is nevertheless restated here in order to review the concepts and standardize the notation. For other basic terminology, the reader may consult the introduction to this book.

• Sections 1.1– 1.2 begin with a brief introduction to subobjects and quotient objects, including the general properties of monomorphisms and epimorphisms. They then introduce the image \(\operatorname{im}(f)\) and coimage \(\operatorname{coim}(f)\) of a morphism \(f\), as well as morphisms for which “image equals coimage,” called strict morphisms. These are basic notions that make sense in arbitrary categories.

• Sections 1.3– 1.4 form the linear part of the chapter. They introduce a special kind of category in which every morphism set has the structure of an abelian group, called an \(\ensuremath{\mathsf{Ab}}\)-category; a functor that preserves addition of morphisms is called additive. In such categories one can define kernels and cokernels of morphisms, generalizing those of module homomorphisms. An \(\ensuremath{\mathsf{Ab}}\)-category with a zero object and finite products is called an additive category. In an additive category one can form finite direct sums of objects, and morphisms between finite direct sums can be manipulated in much the same way as matrices.

  More generally, the additive structure can be enhanced with scalar multiplication by a commutative ring \(\Bbbk\), leading to the notions of a \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category and a \(\Bbbk\)-linear category. Categories with these linear structures are a principal concern of this book.

• Sections 1.5– 1.6 cover the foundations of limits. In particular, Definition 1.5.2 introduces the terminology for a functor \(F:\mathcal C\to\mathcal D\) that preserves limits (from \(\mathcal C\) to \(\mathcal D\)) or creates them (from \(\mathcal D\) to \(\mathcal C\)). Cofinal functors and filtered inductive limits are also discussed; these are useful and important notions.

• Left and right Kan extensions, discussed in Sections 1.7– 1.8, are fundamental operations in category theory. Under suitable conditions they can be constructed using \(\varinjlim\) or \(\varprojlim\). The argument is somewhat involved, but the underlying idea is simple: express the problem of extending a functor as a “best approximation” formulated in terms of limits. We also record the non-obvious Theorem 1.7.7, which explains the connection between absolute left or right Kan extensions and adjoint pairs.

• Gabriel–Zisman localization, introduced in Section 1.9 and henceforth called simply localization, is the construction that formally adjoins inverses to a family \(S\) of morphisms in a category. It is analogous to localization of rings, but the concrete details are considerably more complicated. To retain better control over the resulting category, in practice \(S\) is often assumed to be a multiplicative system in the sense of Definition 1.9.5. The principal use of localization in this book is to construct derived categories; another is to define Serre quotients of abelian categories in Section 2.9. Section 1.10 explains how to form Kan extensions along a localization of categories. This may be viewed as the localization of a functor and is essential for understanding derived functors.

• The adjoint functor theorem discussed in Section 1.11 characterizes, under suitable conditions, when a functor has a left or right adjoint. Its principal application in this book is to the study of Grothendieck categories in Section 2.10. The theorem has uses far beyond this application, so the section is valuable in its own right. The notions of generator and cogenerator discussed there (Definition 1.11.8) are also essential.

This chapter draws extensively on [KS06, Rie16].

Reading Guide. The basic theory of abelian categories and complexes requires only Sections 1.1– 1.4 of this chapter. Sections 1.5– 1.6 provide supporting material and are also cited repeatedly in other chapters. The theory of derived categories uses Sections 1.7– 1.10. This material has applications beyond that theory, but readers may judge their own needs and postpone it until just before studying derived categories. The final section, Section 1.11, is used only in Section 2.10; nevertheless, its material on generators and cogenerators is essential and may be consulted as needed.

Subquotients

We begin by recalling the notions of monomorphism and epimorphism. Fix a category \(\mathcal C\).

• A morphism \(f:X\to Y\) is called a monomorphism, also written \(f:X\hookrightarrow Y\), if left cancellation holds: \[\forall T \in \operatorname{Ob}(\mathcal{C}), \; \forall g,h: T \to X , \quad fg=fh \iff g=h;\]

• A morphism \(f:X\to Y\) is called an epimorphism, also written \(f:X\twoheadrightarrow Y\), if right cancellation holds: \[\forall T \in \operatorname{Ob}(\mathcal{C}), \; \forall g,h: Y \to T, \quad gf=hf \iff g=h.\]

It follows immediately that \(f:X\to Y\) is monic if and only if \(f_*:\operatorname{Hom}(T,X)\to\operatorname{Hom}(T,Y)\) is injective for every \(T\). Likewise, \(f\) is epic if and only if \(f^*:\operatorname{Hom}(Y,T)\to\operatorname{Hom}(X,T)\) is injective for every \(T\). Monicity and epicity are dual properties: \(f\) is monic if and only if \(f^{\operatorname{op}}\) is epic.

Example. If the equalizer \(\operatorname{ker}(f,g)\) of a pair of morphisms \(f,g:X\to Y\) exists, its universal property shows that the canonical morphism \(\iota:\operatorname{ker}(f,g)\to X\) is monic. Dually, if the coequalizer \(\operatorname{coker}(f,g)\) exists, the canonical morphism \(p:Y\to\operatorname{coker}(f,g)\) is epic. Here is another simple observation: if \(\mathcal C\) has a zero object, denoted by \(0\), then \(0\to X\) is monic and \(X\to0\) is epic. The reader is invited to verify these details.

Proposition. Consider morphisms \(X\xrightarrow{a}Y\xrightarrow{b}Z\). Suppose that \(b\) is monic or that \(a\) is epic. Then \(ba:X\to Z\) is an isomorphism if and only if both \(a\) and \(b\) are isomorphisms.

Proof. One direction is clear. Now suppose that \(ba\) is an isomorphism and that \(b\) is monic. Set \(c:=a(ba)^{-1}:Z\to Y\); then \(bc=\ensuremath{\mathrm{id}}_Z\). Moreover, \(b(cb)=(bc)b=b\), so left cancellation by \(b\) gives \(cb=\ensuremath{\mathrm{id}}_Y\). Thus \(b\) is an isomorphism, and consequently so is \(a\). Passing to \(\mathcal C^{\operatorname{op}}\) gives the case in which \(a\) is epic.

Definition.

Let \(\mathcal C\) be a category and let \(X\) be an object of \(\mathcal C\). An equivalence class, under the relation above, of monomorphisms \(S\hookrightarrow X\) (respectively, of epimorphisms \(X\twoheadrightarrow Q\)) is called a subobject (respectively, a quotient object) of \(X\). These equivalence classes form the partially ordered set \((\mathrm{Sub}_X,\subset)\) (respectively, \((\mathrm{Quot}_X,\twoheadleftarrow)\)), in which \(\ensuremath{\mathrm{id}}_X:X\to X\) gives the unique maximal element.

Subobjects and quotient objects are dual: \((\mathrm{Sub}_X,\subset)\) as defined in \(\mathcal C\) is the same as \((\mathrm{Quot}_X,\twoheadleftarrow)\) as defined in \(\mathcal C^{\operatorname{op}}\). For convenience, this section deals mainly with subobjects. Because the isomorphism \(g\) occurring in the equivalence relation is unique, one may choose a representative \(f:S\hookrightarrow X\) of an equivalence class without ambiguity, or abbreviate that representative simply as \(S\). Recall that \(f_*:\operatorname{Hom}(\cdot,S)\to\operatorname{Hom}(\cdot,X)\) is injective.

Lemma. For any two subobjects \(S_1,S_2\) of \(X\), one has \[\begin{equation*} S_1 \subset S_2 \iff \forall T \in \operatorname{Ob}(\mathcal{C}), \; \operatorname{Hom}(T, S_1) \subset \operatorname{Hom}(T, S_2), \end{equation*}\] where \(\operatorname{Hom}(T,S_1)\) and \(\operatorname{Hom}(T,S_2)\) on the right-hand side are understood as subsets of \(\operatorname{Hom}(T,X)\).

Proof. The implication \(\implies\) is immediate. Conversely, take \(T:=S_1\). Under \(\operatorname{Hom}(S_1,S_1)\subset\operatorname{Hom}(S_1,X)\), the element \(\ensuremath{\mathrm{id}}_{S_1}\) corresponds to \(f_1\). The hypothesis supplies \(g\in\operatorname{Hom}(S_1,S_2)\) whose image in \(\operatorname{Hom}(S_1,X)\) is \(f_1\), that is, \(f_2g=f_1\). This is the required morphism.

Suppose that \(\mathcal C\) has a zero object and that \((X_i)_{i\in I}\) is a family of objects. For every \((i,j)\in I\times I\), define \(\delta_{ij}\in\operatorname{Hom}(X_i,X_j)\) by \(\delta_{ij}=\ensuremath{\mathrm{id}}\) when \(i=j\), and as the zero morphism otherwise. If the product and coproduct of \((X_i)_{i\in I}\) exist, then \((\delta_{ij})_{i,j}\) determines a canonical morphism \[\begin{equation} \delta: \coprod_{i \in I} X_i \to \prod_{i \in I} X_i. \end{equation}\] For the additive categories to be reviewed shortly, \(\delta\) is always an isomorphism when \(I\) is finite. This need not hold in an arbitrary category.

Lemma. Let \(X\to Z\) be a monomorphism. Its pullback along \(Y\to Z\), namely \(X\ensuremath{\underset{Z}{\times}}Y\to Y\) (if it exists), is also a monomorphism. Dually, a pushout of an epimorphism is again an epimorphism.

Lemma. Let \((f_i:X_i\to Z)_{i\in I}\) be a family of monomorphisms. Then the canonical morphism from \(\prod_{i\in I}(X_i\to Z)\) (if it exists) to \(Z\) is also a monomorphism. Dually, given a family of epimorphisms \((g_i:Z\to X_i)_{i\in I}\), the canonical morphism from \(Z\) to \(\coprod_{i\in I}(Z\to X_i)\) (if it exists) is also an epimorphism.

We return to subobjects and quotient objects.

Definition.

Let \(\mathcal C\) be any category and let \(X,Y\) be objects of \(\mathcal C\). If \(Y\) can be realized as a quotient object of a subobject of \(X\), then \(Y\) is called a subquotient of \(X\).

Considering a subobject of a quotient object amounts to discussing a subquotient in \(\mathcal C^{\operatorname{op}}\). The next result shows that, for a certain class of categories, the two approaches yield the same notion. This class includes the abelian categories studied later; see Corollary 2.1.7.

Proposition. Suppose that \(\mathcal C\) has finite fiber products and finite fiber coproducts, that pullbacks of epimorphisms remain epimorphisms, and that pushouts of monomorphisms remain monomorphisms. Then a subquotient may equivalently be defined as a subobject of a quotient object.

Proof. Consider a subquotient diagram \(Y\twoheadleftarrow Z\hookrightarrow X\) in \(\mathcal C\), and set \(W:=X\ensuremath{\underset{Z}{\sqcup}}Y\). The morphism \(X\to W\) is the pushout of \(Z\to Y\), so it is epic by Lemma reference; the morphism \(Y\to W\) is the pushout of \(Z\to X\), so it is monic by hypothesis. Thus \(Y\hookrightarrow W\twoheadleftarrow X\), exhibiting a subquotient of \(X\) in \(\mathcal C^{\operatorname{op}}\). Since the assumptions are self-dual, a subquotient in \(\mathcal C^{\operatorname{op}}\) likewise yields a subquotient in \(\mathcal C\).

Once the property “quotient of a subobject = subobject of a quotient” has been established, it follows that a subquotient of a subquotient is again a subquotient. The definition of a subquotient does not specify how it is to be realized as a quotient object of a subobject. For example, in \(\mathcal C=\ensuremath{\mathsf{Ab}}\), \(\ensuremath{\mathbb{Z}}/2\ensuremath{\mathbb{Z}}\) is a subquotient of \(\ensuremath{\mathbb{Z}}/4\ensuremath{\mathbb{Z}}\): it can be realized either as a quotient object of \(\ensuremath{\mathbb{Z}}/4\ensuremath{\mathbb{Z}}\) or as the subobject \(2\ensuremath{\mathbb{Z}}/4\ensuremath{\mathbb{Z}}\) of \(\ensuremath{\mathbb{Z}}/4\ensuremath{\mathbb{Z}}\).

Images, Coimages, and Strict Morphisms

Let \(\mathcal C\) be a category. For any morphism \(f:X\to Y\), the fiber coproduct \(Y\ensuremath{\underset{X}{\sqcup}}Y\), if it exists, comes equipped with a canonical pair \(Y\rightrightarrows Y\ensuremath{\underset{X}{\sqcup}}Y\). Dually, the fiber product \(X\ensuremath{\underset{Y}{\times}}X\), if it exists, comes equipped with a canonical pair \(X\ensuremath{\underset{Y}{\times}}X\rightrightarrows X\), usually called the projection morphisms.

Definition (Image and Coimage).

Let \(f:X\to Y\) be a morphism in \(\mathcal C\). Suppose that \(Y\ensuremath{\underset{X}{\sqcup}}Y\) exists. If the equalizer \(\operatorname{ker}\left(Y\rightrightarrows Y\ensuremath{\underset{X}{\sqcup}}Y\right)\) exists, this equalizer is called the image of \(f\) and denoted by \(\operatorname{im}(f)\); it carries a canonical monomorphism \(\operatorname{im}(f)\hookrightarrow Y\).

Dually, suppose that \(X\ensuremath{\underset{Y}{\times}}X\) exists. If the coequalizer \(\operatorname{coker}\left(X\ensuremath{\underset{Y}{\times}}X\rightrightarrows X\right)\) exists, this coequalizer is called the coimage of \(f\) and denoted by \(\operatorname{coim}(f)\); it carries a canonical epimorphism \(X\twoheadrightarrow\operatorname{coim}(f)\).

Remark. The duality between \(\operatorname{ker}\) and \(\operatorname{coker}\) immediately shows that \(\operatorname{im}(f)\) exists in \(\mathcal C\) if and only if \(\operatorname{coim}(f^{\operatorname{op}})\) exists in \(\mathcal C^{\operatorname{op}}\), and then \(\operatorname{im}(f)=\operatorname{coim}(f^{\operatorname{op}})\).

Besides being useful, the following lemma sheds light on the nature of images and coimages.

Proof. The two versions are dual, so it suffices to consider the one involving \(j\). Let \(p_1,p_2\) be the projections \(X\ensuremath{\underset{Y}{\times}}X\rightrightarrows X\). Since \(jgp_1=fp_1=fp_2=jgp_2\) and \(j\) is monic, \(gp_1=gp_2\). The universal property of \(\operatorname{coim}(f)\) as a coequalizer then uniquely determines \(\overline g\) making the diagram commute.

Definition (Strict morphism). Suppose that \(\operatorname{im}(f)\) and \(\operatorname{coim}(f)\) exist. If the morphism \(\operatorname{coim}(f)\to\operatorname{im}(f)\) in equation is an isomorphism, then \(f\) is called strict.

For example, every morphism in \(\mathcal C=\ensuremath{\mathsf{Set}}\) is strict, and both its image and its coimage are canonically identified with its set-theoretic image. Further examples appear in the exercises for this chapter.

Lemma. For a morphism \(f:X\to Y\) in \(\mathcal C\), \[f\;\text{epic} \iff \operatorname{im}(f) \ensuremath{\stackrel{\sim}{\rightarrow}}Y, \quad f\;\text{monic} \iff X \ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{coim}(f).\]

Proof. By duality it suffices to prove the first equivalence. If \(f\) is epic, the canonical morphisms \(i_1,i_2:Y\rightrightarrows Y\ensuremath{\underset{X}{\sqcup}}Y\) satisfy \(i_1f=i_2f\), hence \(i_1=i_2\), and therefore \(\operatorname{im}(f):=\operatorname{ker}(i_1,i_2)=Y\).

Conversely, suppose the canonical morphism \(\operatorname{im}(f)\to Y\) is an isomorphism. Since composing it with \(i_1\) and \(i_2\) gives the same morphism, again \(i_1=i_2\). If \(Z\in\operatorname{Ob}(\mathcal C)\) and \(g_1,g_2:Y\rightrightarrows Z\) satisfy \(g_1f=g_2f\), the universal property yields \(g:Y\ensuremath{\underset{X}{\sqcup}}Y\to Z\) with \(g_j=gi_j\) for \(j=1,2\). Hence \(g_1=g_2\), proving that \(f\) is epic.

Proposition. A strict morphism \(f\) is an isomorphism if and only if it is both monic and epic.

Proof. If \(f\) is strict, monic, and epic, factor it as \(X\twoheadrightarrow\operatorname{coim}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}(f)\hookrightarrow Y\) and apply Proposition reference and Lemma reference.

The preceding arguments involve diagrams \(X\twoheadrightarrow C\hookrightarrow Y\) whose composite is \(f\). Such a diagram is an epi–mono factorization of \(f:X\to Y\). In practice one studies not only the factorizations themselves, but also the morphisms between them.

1. If a \(\varphi\) as shown by the dashed arrow makes the left half (respectively right half) commute, then the other half also commutes. Such a \(\varphi\), if it exists, is unique.

2. If \(a\) is epic (respectively \(b\) is monic) and \(\varphi\) exists, then \(\varphi\) is epic (respectively monic).

Proof. Suppose that \(\varphi:C\to C'\) makes the left half commute and that \(e,e'\) are epic. The equation \(\varphi e=e'a\), together with the epicity of \(e\), uniquely determines \(\varphi\). Since the outer rectangle commutes, \(m'\varphi e=m'e'a=bme\); cancelling \(e\) then gives \(m'\varphi=bm\), so the right half commutes. If \(a\) is also epic, then \(\varphi e=e'a\) is epic, and hence \(\varphi\) is epic.

This proves one half of (i) and (ii). The case where \(\varphi\) makes the right half commute and \(m,m'\) are monic is analogous, or equivalently dual.

Taking \(a=\ensuremath{\mathrm{id}}_X\) and \(b=\ensuremath{\mathrm{id}}_Y\) in Lemma reference shows that the \(\varphi\) above, if it exists, is unique and both epic and monic. Thus \(\ensuremath{\mathsf{epi.mono}}(f)\) is the category associated with a preordered set. This section considers only one special case of epi–mono factorizations.

Proposition. Suppose that every morphism in \(\mathcal C\) is strict. Then every \(f:X\to Y\) has an epi–mono factorization \(X\twoheadrightarrow C\hookrightarrow Y\), unique up to isomorphism in \(\ensuremath{\mathsf{epi.mono}}(f)\).

Inferring that a subdiagram commutes from the commutativity of its outer frame is a standard technique that will be used repeatedly.

Additive Categories: Kernels and Cokernels

This section uses the notion of a functor preserving \(\varinjlim\) or \(\varprojlim\). This is basic category theory; see [Li1] or the brief review in Section 1.5. We first recall \(\ensuremath{\mathsf{Ab}}\)-categories and additive categories; see [Li1].

• A category \(\mathcal C\) is an \(\ensuremath{\mathsf{Ab}}\)-category, or a preadditive category, if every \(\operatorname{Hom}(X,Y)\) is an abelian group, written additively (equivalently, a \(\ensuremath{\mathbb{Z}}\)-module), and every composition map \(\operatorname{Hom}(Y,Z)\times\operatorname{Hom}(X,Y)\to\operatorname{Hom}(X,Z)\) is \(\ensuremath{\mathbb{Z}}\)-bilinear. One may therefore speak of the morphism \(0\in\operatorname{Hom}(X,Y)\). An \(\ensuremath{\mathsf{Ab}}\)-category may have \(\ensuremath{\mathsf{Ab}}\)-subcategories; every full subcategory is automatically one.

• If \(\mathcal C_1,\mathcal C_2\) are \(\ensuremath{\mathsf{Ab}}\)-categories, a functor \(F:\mathcal C_1\to\mathcal C_2\) is additive if the induced maps on \(\operatorname{Hom}\) sets are group homomorphisms. Equivalences of \(\ensuremath{\mathsf{Ab}}\)-categories and their quasi-inverses are always understood to be additive.

We shall need the following elementary but fundamental observation. By convention, a linear map of \(\ensuremath{\mathbb{Z}}\)-modules means a module homomorphism.

Proposition. Let \(\mathcal C\) and \(\mathcal C'\) be \(\ensuremath{\mathsf{Ab}}\)-categories.

1. Given adjoint functors \[\text{Diagram chapter1-unit-011-diagram-001},\] with \(F,G\) additive, the adjunction isomorphism \(\varphi:\operatorname{Hom}_{\mathcal C'}(F(\cdot),\cdot)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{Hom}_{\mathcal C}(\cdot,G(\cdot))\) is linear.

2. For any \(\alpha:I\to\mathcal C\), the isomorphism in the universal property \[\operatorname{Hom}_{\mathcal{C}}\left( \varinjlim \alpha, T \right) \simeq \varprojlim_{i \in \operatorname{Ob}(I)} \operatorname{Hom}_{\mathcal{C}}(\alpha(i), T), \quad T \in \operatorname{Ob}(\mathcal{C})\] is linear whenever the colimit exists. The analogous assertion holds for \(\varprojlim\).

Proof. For (i), the discussion following [Li1] says that \(\varphi\) is determined by the unit \(\eta\) and counit \(\varepsilon\): \[\varphi(f) = Gf \circ \eta_X, \quad \varphi^{-1}(g) = \varepsilon_Y \circ Fg, \quad f \in \operatorname{Hom}_{\mathcal{C}'}(FX, Y), \; g \in \operatorname{Hom}_{\mathcal{C}}(X, GY);\] bilinearity of composition makes \(\varphi\) linear. For (ii), the isomorphism is described by composition with \(\iota_i:\alpha(i)\to\varinjlim\alpha\), so it too is linear.

In an \(\ensuremath{\mathsf{Ab}}\)-category, \(X_1\times X_2\) exists if and only if \(X_1\sqcup X_2\) exists, and then there is a canonical isomorphism \(X_1\sqcup X_2\ensuremath{\stackrel{\sim}{\rightarrow}}X_1\times X_2\). This structure is called a biproduct; see [Li1].

Convention.

Write the \(n\)-fold biproduct \(Z\) as \(X_1\oplus\cdots\oplus X_n\). If binary biproducts exist, all \(n\)-fold biproducts for \(n\geq1\) can be formed iteratively using the canonical isomorphism \(X_1\oplus X_2\oplus X_3\simeq(X_1\oplus X_2)\oplus X_3\). Also set \[X^{\oplus n} := \underbracket{X \oplus \cdots \oplus X}_{n\; \text{terms}}.\] The product diagonal \(\delta_X:X\to X^{\oplus n}\) and the coproduct codiagonal \(\check\delta_X:X^{\oplus n}\to X\) are characterized by \(p_i\delta_X=\ensuremath{\mathrm{id}}_X=\check\delta_X\iota_i\) for \(1\leq i\leq n\). Explicitly, \(\delta_X=\sum_{i=1}^n\iota_i\) and \(\check\delta_X=\sum_{i=1}^np_i\).

Definition (Additive category).

An \(\ensuremath{\mathsf{Ab}}\)-category \(\mathcal C\) is additive if:

• it has a zero object \(0\);

• every two objects \(X,Y\) have a biproduct \(X\oplus Y\).

An \(\ensuremath{\mathsf{Ab}}\)-subcategory of an additive category that is itself additive is an additive subcategory. Every finite family \(X_1,\ldots,X_n\) in an additive category has a biproduct \(X_1\oplus\cdots\oplus X_n\), also called their direct sum; the \(X_i\) are its direct summands.

An additive functor \(F:\mathcal C_1\to\mathcal C_2\) between additive categories preserves all finite products and coproducts, because:

• \(F\) preserves biproducts; see [Li1];

• \(F\) preserves zero objects, since \(F(\ensuremath{\mathrm{id}}_X)=\ensuremath{\mathrm{id}}_{FX}\) and a zero object is characterized by \(\ensuremath{\mathrm{id}}=0\); see [Li1].

If \(\mathcal C\) is an \(\ensuremath{\mathsf{Ab}}\)-category (respectively additive category), then \(\mathcal C^{\operatorname{op}}\) is naturally one as well, with \(f\mapsto f^{\operatorname{op}}\) an isomorphism \(\operatorname{Hom}_{\mathcal C^{\operatorname{op}}}(Y,X)\to\operatorname{Hom}_{\mathcal C}(X,Y)\) of groups. Henceforth, without ambiguity, \(0\) denotes both zero objects and zero morphisms in additive categories. A family \(f_i:X_i\to X'_i\), \(i=1,\ldots,n\), naturally induces \(f_1\oplus\cdots\oplus f_n:\bigoplus_iX_i\to\bigoplus_iX'_i\), explicitly \(\sum_{i=1}^n\iota'_if_ip_i\).

Example. The category \(\ensuremath{\mathsf{Ab}}\) is additive: addition on \(\operatorname{Hom}\) sets is pointwise, \((f+g)(x)=f(x)+g(x)\); the zero object is the zero group; and biproducts are the usual direct sums. Similarly, \(R\ensuremath{\text{-}\mathsf{Mod}}\) is additive for any ring \(R\), and the forgetful functor \(R\ensuremath{\text{-}\mathsf{Mod}}\to\ensuremath{\mathsf{Ab}}\) is additive. The category \(\ensuremath{\mathsf{Ban}}_{\ensuremath{\mathbb{C}}}\) of complex Banach spaces and continuous linear maps is also additive. Addition is again pointwise and the zero object is the zero space. For Banach spaces \((X_i,\|\cdot\|_i)\), \(i=1,2\), the biproduct may be taken to be the vector-space direct sum \(X_1\oplus X_2\) equipped with the norm \(\|(x_1,x_2)\|:=\max\{\|x_1\|_1,\|x_2\|_2\}\), where \(x_i\in X_i\).

Corollary. Let \(\mathcal C,\mathcal C'\) be additive categories and \(F:\mathcal C\to\mathcal C'\) a functor.

1. \(F\) preserves finite products if and only if it preserves finite coproducts.

2. If \(F\) preserves finite products, equivalently finite coproducts, then it is additive.

3. If \(F\) preserves zero objects and biproducts, then it is additive.

4. If \(F\) is an equivalence, then \(F\) and its quasi-inverse are additive.

5. If \[\text{Diagram chapter1-unit-011-d004}\] is an adjoint pair, then \(F,G\) are additive; Proposition reference therefore applies.

In fact, Proposition reference shows that “additivity” is a property of the underlying category \(\mathcal C\), not additional structure.

Definition.

Let \(\mathcal C\) be an \(\ensuremath{\mathsf{Ab}}\)-category with a zero object, and \(f:X\to Y\) a morphism.

• If the equalizer \(\operatorname{ker}(f,0)\) exists, then \(\operatorname{ker}(f):=\operatorname{ker}(f,0)\) together with \(\operatorname{ker}(f)\hookrightarrow X\) is the kernel of \(f\).

• If the coequalizer \(\operatorname{coker}(f,0)\) exists, then \(\operatorname{coker}(f):=\operatorname{coker}(f,0)\) together with \(Y\twoheadrightarrow\operatorname{coker}(f)\) is the cokernel of \(f\).

Thus the composites \(\operatorname{ker}(f)\to X\to Y\) and \(X\to Y\to\operatorname{coker}(f)\) are both \(0\).

Kernels and cokernels admit several complementary viewpoints.

• For a morphism \(g:T\to X\) satisfying \(fg=0\),¹ the universal property of \(\operatorname{ker}(f)=\operatorname{ker}(f,0)\) is the commutative diagram \[\text{Diagram chapter1-unit-011-d006}\]

• For a morphism \(h:Y\to T\) satisfying \(hf=0\), the universal property of \(\operatorname{coker}(f)=\operatorname{coker}(f,0)\) is \[\text{Diagram chapter1-unit-011-d007}\]

• The constructions are dual: \(\operatorname{coker}(f)=\operatorname{ker}(f^{\operatorname{op}})\) and \(\operatorname{ker}(f)=\operatorname{coker}(f^{\operatorname{op}})\).

From now on, \(\mathcal C\) will mainly be additive.

Proof. By equation, iterated fiber products give canonical isomorphisms \(\operatorname{ker}(gf)\simeq X\ensuremath{\underset{Z}{\times}}0\simeq X\ensuremath{\underset{Y}{\times}}(Y\ensuremath{\underset{Z}{\times}}0)\simeq X\ensuremath{\underset{Y}{\times}}\operatorname{ker}(g)\). Duality gives the cokernel statement.

Proof. By duality it suffices to treat pullbacks. For every \(T\), the universal properties give canonical isomorphisms \[\begin{align*} \operatorname{Hom}(T, \operatorname{ker}(f)) & \simeq \operatorname{ker}\left[ \operatorname{Hom}(T, X) \to \operatorname{Hom}(T, Y) \right] \\ & \simeq \operatorname{ker}\left[ \operatorname{Hom}(T, Y) \ensuremath{\underset{\operatorname{Hom}(T, Y')}{\times}} \operatorname{Hom}(T, X') \to \operatorname{Hom}(T, Y) \right] \\ & = \operatorname{ker}\left[ \operatorname{Hom}(T, X') \to \operatorname{Hom}(T, Y') \right] \simeq \operatorname{Hom}(T, \operatorname{ker}(g)), \end{align*}\] and the Yoneda lemma gives \(\operatorname{ker}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{ker}(g)\); see Theorem A.1.1.

Lemma. Let \(\mathcal C\) be additive and \(f:X\to Y\) a morphism.

1. \(f\) is monic (respectively epic) if and only if \(\operatorname{ker}(f)=0\) (respectively \(\operatorname{coker}(f)=0\)).

2. \(f\) is zero if and only if \(\operatorname{ker}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}X\), equivalently \(Y\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{coker}(f)\).

3. Given \(k:Y\to Z\) (respectively \(k':W\to X\)), there is a unique morphism making the corresponding diagram commute: \[\text{Diagram chapter1-unit-011-d016} \quad \text{or} \quad \text{Diagram chapter1-unit-011-d017}\] provided the indicated kernels or cokernels exist. If \(k\) is monic (respectively \(k'\) is epic), then \(\operatorname{ker}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{ker}(kf)\) (respectively \(\operatorname{coker}(fk')\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{coker}(f)\)).

In an additive category, the image and coimage of Definition reference have simple descriptions.

Proposition.

For \(f:X\to Y\) in an additive category \(\mathcal C\), there are canonical isomorphisms \[\begin{align*} \operatorname{im}(f) & \simeq \operatorname{ker}\left[ Y \to \operatorname{coker}(f) \right] \hookrightarrow Y, \\ \operatorname{coim}(f) & \simeq \operatorname{coker}\left[ \operatorname{ker}(f) \to X \right] \twoheadleftarrow X ; \end{align*}\] provided the indicated kernels and cokernels exist.

Proof. By duality consider only \(\operatorname{im}(f)\). Let \(i_1,i_2:Y\to Y\ensuremath{\underset{X}{\sqcup}}Y\) be canonical. By Yoneda it suffices to verify, for every \(T\), \[\begin{multline*} \operatorname{Hom}\left( T, \operatorname{im}(f) \right) = \left\{ h: T \to Y \;\middle|\; i_1 h = i_2 h \right\} \\ = \left\{ \begin{array}{r|l} h: T \to Y & \forall S \in \operatorname{Ob}(\mathcal{C}),\; \forall s_1, s_2: Y \to S, \\ & s_1 f = s_2 f \implies s_1 h = s_2 h \end{array}\right\} \\ \xlongequal{s := s_1 - s_2} \left\{ \begin{array}{r|l} h: T \to Y & \forall S \in \operatorname{Ob}(\mathcal{C}),\; \forall s: Y \to S, \\ & s f = 0 \implies sh = 0 \end{array}\right\} \\ = \left\{ \begin{array}{r|l} h: T \to Y & T \xrightarrow{h} Y \twoheadrightarrow \operatorname{coker}(f) \\ & \text{has composite}\; 0 \end{array}\right\} \ensuremath{\stackrel{\sim}{\leftarrow}}\operatorname{Hom}\left( T, \operatorname{ker}\left[ Y \to \operatorname{coker}(f) \right] \right). \end{multline*}\] The first equality is \(\operatorname{im}(f):=\operatorname{ker}(i_1,i_2)\). For the second, rewrite the condition as \(s'i_1h=s'i_2h\) for every \(s':Y\ensuremath{\underset{X}{\sqcup}}Y\to S\) and use \[\begin{align*} \operatorname{Hom}\left( Y \ensuremath{\underset{X}{\sqcup}} Y, S \right) & \xrightarrow{1:1} \operatorname{Hom}(Y,S) \ensuremath{\underset{\operatorname{Hom}(X,S)}{\times}} \operatorname{Hom}(Y,S) \\ s' & \longmapsto \left( s' i_1, s' i_2 \right) =: (s_1, s_2). \end{align*}\] The fourth equality follows from the cokernel’s universal property, reducing to \(S=\operatorname{coker}(f)\).

If \(f=0\), Lemma reference(ii) gives the simplest example: \(\operatorname{im}(f)=0=\operatorname{coim}(f)\).

Example. Continuing Example reference, consider \(R\ensuremath{\text{-}\mathsf{Mod}}\). A module homomorphism \(f:A\to B\) always has \(\operatorname{ker}(f)=f^{-1}(0)\) and \(\operatorname{coker}(f)=B/\{f(a):a\in A\}\). Its \(\operatorname{im}(f)\) is the set-theoretic image, \(\operatorname{coim}(f)=A/\operatorname{ker}(f)\), and the comparison in equation is the canonical module isomorphism \(A/\operatorname{ker}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}(f)\). Thus every morphism in \(R\ensuremath{\text{-}\mathsf{Mod}}\) is strict.

---

1. Translator’s note: the source says “for every morphism” without including the condition \(fg=0\), or dually \(hf=0\). These conditions have been restored because they are precisely the equalizer and coequalizer hypotheses; see correction O014-C006.↩︎

2. Translator’s note: the source says “coproduct,” but the displayed object is the fiber coproduct \(Z\ensuremath{\underset{Y}{\sqcup}}\operatorname{coker}(f)\) and the proof uses duality with the fiber product; see correction O014-C005.↩︎

Generalization: Linear Categories over a Commutative Ring

The \(\operatorname{Hom}\) sets of an additive category have the structure of abelian groups. In many situations they also carry scalar multiplication by some commutative ring \(\Bbbk\). For example, every \(\operatorname{Hom}\) set in \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\) is naturally a \(\Bbbk\)-module, and composition of morphisms is \(\Bbbk\)-bilinear.

Definition.

Let \(F: \mathcal{A} \to \mathcal{A}'\) be a functor between \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-categories. If the map \(\operatorname{Hom}(X,Y) \to \operatorname{Hom}(FX,FY)\) is a \(\Bbbk\)-module homomorphism for every \(X,Y \in \operatorname{Ob}(\mathcal{A})\), then \(F\) is called a \(\Bbbk\)-linear functor.

Observe that \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\) is a monoidal category under the tensor product \(\otimes := \ensuremath{\underset{\Bbbk}{\otimes}}\). Thus the term “\(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category” agrees with the terminology for enriched categories in [Li1]. The \(\ensuremath{\mathsf{Ab}}\)-categories introduced earlier are precisely the \(\ensuremath{\mathbb{Z}}\ensuremath{\text{-}\mathsf{Mod}}\)-categories. Conversely, forgetting scalar multiplication on a \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category leaves an \(\ensuremath{\mathsf{Ab}}\)-category.

Example. A \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category with a single object is nothing but a \(\Bbbk\)-algebra. The correspondence is given by \(\mathcal{A} \mapsto \operatorname{End}_{\mathcal{A}}(\star)\), where \(\operatorname{Ob}(\mathcal{A}) = \{\star\}\).

The argument of Proposition reference carries over verbatim and gives the following result.

Proposition. Let \(\mathcal{A}\) and \(\mathcal{A}'\) be \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-categories.

1. Given a pair of adjoint functors

   \[\text{Diagram chapter1-unit-012-d002},\] with both \(F\) and \(G\) \(\Bbbk\)-linear, the adjunction isomorphism \(\operatorname{Hom}_{\mathcal{A}'}(F(\cdot),\cdot) \ensuremath{\stackrel{\sim}{\rightarrow}} \operatorname{Hom}_{\mathcal{A}}(\cdot,G(\cdot))\) is \(\Bbbk\)-linear.

2. For every \(\alpha: I \to \mathcal{A}\), the bijection \[\operatorname{Hom}_{\mathcal{A}}\left(\varinjlim \alpha,T\right) \simeq \varprojlim_{i \in \operatorname{Ob}(I)} \operatorname{Hom}_{\mathcal{A}}(\alpha(i),T), \quad T \in \operatorname{Ob}(\mathcal{A})\] is \(\Bbbk\)-linear whenever \(\varinjlim \alpha\) exists. The same statement holds for \(\varprojlim\).

Unless otherwise specified, all functors and equivalences between \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-categories will henceforth be understood to be \(\Bbbk\)-linear.

Definition.

If an additive category \(\mathcal{A}\) is equipped with a \(\Bbbk\)-linear structure compatible with its existing \(\ensuremath{\mathsf{Ab}}\)-category structure, then \(\mathcal{A}\) together with these data is called a \(\Bbbk\)-linear category.

This amounts to replacing the \(\ensuremath{\mathsf{Ab}}\)-category by a \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category in the discussion of §reference. Most results about additive categories in this chapter extend to the \(\Bbbk\)-linear setting by exactly the same arguments. The sole exception concerns Corollary reference: in an additive category, addition on the \(\operatorname{Hom}\) sets can be characterized by properties of the category itself, but this is not true of scalar multiplication by \(\Bbbk\). Consequently, functors—and even equivalences—between \(\Bbbk\)-linear categories are not automatically \(\Bbbk\)-linear. We next collect a few results concerning \(\Bbbk\)-linearity. Recall that the center of a category \(\mathcal{A}\) is defined by

\(Z(\mathcal{A}) := \operatorname{End}(\ensuremath{\mathrm{id}}_{\mathcal{A}})\). Its elements are endomorphisms of the identity functor \(\ensuremath{\mathrm{id}}_{\mathcal{A}}\), written as \((a_X)_{X \in \operatorname{Ob}(\mathcal{A})}\) with \(a_X \in \operatorname{End}_{\mathcal{A}}(X)\). Composition makes the center a commutative monoid; see [Li1]. If \(\mathcal{A}\) is an \(\ensuremath{\mathsf{Ab}}\)-category, then \(Z(\mathcal{A})\) becomes a commutative ring under composition and addition \((a_X)_X + (b_X)_X := (a_X+b_X)_X\).

Proposition. Let \(\mathcal{A}\) be an \(\ensuremath{\mathsf{Ab}}\)-category. Then there is a bijection \[\left\{\text{$\Bbbk\ensuremath{\text{-}\mathsf{Mod}}$-category structures on $\mathcal{A}$}\right\} \stackrel{1:1}{\longleftrightarrow} \operatorname{Hom}_{\text{rings}}(\Bbbk,Z(\mathcal{A})).\] The map is given as follows. If a \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category structure on \(\mathcal{A}\) has been fixed, define, for every \(a \in \Bbbk\) and \(X \in \operatorname{Ob}(\mathcal{A})\), \(a_X := a \cdot \ensuremath{\mathrm{id}}_X \in \operatorname{End}_{\mathcal{A}}(X)\). Then \(a \mapsto (a_X)_{X \in \operatorname{Ob}(\mathcal{A})} \in Z(\mathcal{A})\) is a ring homomorphism. Conversely, given a ring homomorphism on the right, \(a \mapsto (a_X)_{X \in \operatorname{Ob}(\mathcal{A})}\), define a \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category structure on \(\mathcal{A}\) by \(a \cdot f := f \circ a_X = a_Y \circ f\), for \(f \in \operatorname{Hom}(X,Y)\).

Proof. The condition for \((a_X)_{X \in \operatorname{Ob}(\mathcal{A})}\) to belong to \(Z(\mathcal{A})\) is precisely \(f \circ a_X = a_Y \circ f\) for every \(f \in \operatorname{Hom}(X,Y)\). If \(\mathcal{A}\) is equipped with a \(\Bbbk\ensuremath{\text{-}\mathsf{Mod}}\)-category structure, define \(a_X := a \cdot \ensuremath{\mathrm{id}}_X\). Bilinearity gives \[f \circ a_X = f \circ (a \cdot \ensuremath{\mathrm{id}}_X) = (a f) \circ \ensuremath{\mathrm{id}}_X = (a \cdot \ensuremath{\mathrm{id}}_Y) \circ f = a_Y \circ f,\] so \((a_X)_{X \in \operatorname{Ob}(\mathcal{A})} \in Z(\mathcal{A})\). The remaining verifications are immediate.

Consequently, the \(\ensuremath{\mathsf{Ab}}\)-category \(\mathcal{A}\) is naturally a \(Z(\mathcal{A})\ensuremath{\text{-}\mathsf{Mod}}\)-category.

Viewing Limits through Functors

There are two broad kinds of question about the relationship between a functor \(F: \mathcal{C} \to \mathcal{D}\) and limits. Taking \(\varinjlim\) as an example, let \(\alpha: I \to \mathcal{C}\) be a functor. One may ask:

• Suppose \(\varinjlim \alpha\) exists. Does its image \(F\left(\varinjlim \alpha\right)\) yield \(\varinjlim (F\alpha)\)?

• Suppose \(\varinjlim F\alpha\) exists. Can this limit be “lifted” to \(\mathcal{C}\)? In what sense is such a lift unique?

We focus on the case of \(\varinjlim\). These results of course have corresponding versions for \(\varprojlim\); by duality, there is no need to repeat the details. We organize the discussion of \(\varinjlim\) in terms of cocones and the category \((\alpha/\Delta)\).

Convention. For a fixed functor \(\alpha: I \to \mathcal{C}\), a collection of data \((L, (f_i)_{i \in \operatorname{Ob}(I)})\) in \(\mathcal{C}\) satisfying the following conditions is called a cocone with base \(\alpha\) and vertex \(L\):

• \(L\) is an object of \(\mathcal{C}\);

• each \(f_i : \alpha(i) \to L\) is a morphism in \(\mathcal{C}\) and satisfies the compatibility condition \(f_j \circ \alpha(i \to j) = f_i\), as \([i \to j]\) ranges over \(\mathrm{Mor}(I)\).

In the evident way, the functor \(F\) induces a functor \((\alpha/\Delta) \to (F\alpha / \Delta)\) that takes a cocone \((L, (f_i)_i)\) to \((FL, (Ff_i)_i)\). Suppose \(\varinjlim \alpha\) exists. If \(\varinjlim F\alpha\) also exists, there is a unique morphism in \((F\alpha / \Delta)\) \[\varinjlim F\alpha \to F(\varinjlim \alpha),\] and this morphism is an isomorphism if and only if \(\left( F(\varinjlim \alpha), (F\iota_i)_i \right)\) gives \(\varinjlim F\alpha\). Dually, \(\varprojlim \alpha\) is characterized as a terminal object of \((\Delta/\alpha)\). Assuming that both projective limits under discussion exist, there is a unique morphism in \((\Delta/F\alpha)\) \[F\varprojlim \alpha \to \varprojlim F\alpha.\]

Definition.

Given \(F: \mathcal{C} \to \mathcal{D}\) and a functor \(\alpha: I \to \mathcal{C}\), consider the induced functor between cocone categories \((\alpha/\Delta) \to (F\alpha/\Delta)\).

• The functor \(F\) is said to preserve this \(\varinjlim\) if \((\alpha/\Delta) \to (F\alpha/\Delta)\) takes an initial object, when one exists, to an initial object.

• The functor \(F\) is said to reflect this \(\varinjlim\) if any object of \((\alpha/\Delta)\) whose image is an initial object of \((F\alpha/\Delta)\) is itself an initial object.

• The functor \(F\) is said to create this \(\varinjlim\) if, whenever \(\varinjlim F\alpha\) exists, \(\varinjlim \alpha\) also exists, and in this situation \(F\) preserves this \(\varinjlim\) and also reflects this \(\varinjlim\).

There are corresponding notions for \(\varprojlim\). The notions for \(\varinjlim\) and \(\varprojlim\) are obtained from one another by replacing \(\mathcal{C}\) and \(\mathcal{D}\) with \(\mathcal{C}^{\operatorname{op}}\) and \(\mathcal{D}^{\operatorname{op}}\).

Notice that these notions depend on the particular \(\alpha\) under consideration. For example, \(F\) may well preserve every finite \(\varinjlim\) without preserving \(\varinjlim\) in general. For a fixed \(\alpha\), saying that a functor creates \(\varinjlim\) (or \(\varprojlim\)) means that the cocone (or cone) in \(\mathcal{D}\) defining the limit can be lifted uniquely to \(\mathcal{C}\), and that the lifted cocone (or cone) gives the corresponding limit in \(\mathcal{C}\). This uniqueness comes from the condition in the definition concerning reflection of \(\varinjlim\) (or \(\varprojlim\)). This point of view will be evident in Example reference and Proposition reference below.

Proposition. Every fully faithful functor reflects every \(\varinjlim\) and every \(\varprojlim\).

Proof. It suffices to consider \(\varinjlim\). If \(F\) is fully faithful, the universal property of an initial object of \((\alpha/\Delta)\) can be checked, after applying \(F\), in \((F\alpha/\Delta)\).

We shall also use the following notion.

Definition.

A functor \(F\) is called conservative if it satisfies the following condition: a morphism \(f \in \operatorname{Mor}(\mathcal{C})\) is an isomorphism if and only if \(Ff \in \operatorname{Mor}(\mathcal{D})\) is an isomorphism.

Remark. Suppose \(F\) is a conservative functor. If \(\varinjlim \alpha\) exists and \(F\) preserves this \(\varinjlim\), the functor \(F\) must also reflect it. Indeed, suppose \(L \in \operatorname{Ob}((\alpha/\Delta))\) is mapped to an initial object of \((F\alpha/\Delta)\). Consider the unique morphism \(\varinjlim \alpha \to L\) in \((\alpha/\Delta)\). Since \(F\) preserves this \(\varinjlim\), the image of that morphism under \(F\) must be an isomorphism. Conservativity then ensures that \(\varinjlim \alpha \to L\) is also an isomorphism.

Under these hypotheses, the condition that \(F\) reflect \(\varinjlim\) may be omitted from the definition of what it means for \(F\) to create \(\varinjlim\).

Example. We verify that the forgetful functor \(U\) from the category of groups \(\ensuremath{\mathsf{Grp}}\) to the category of sets \(\ensuremath{\mathsf{Set}}\) creates all small \(\varprojlim\).

This follows from the concrete constructions of \(\varprojlim\) in \(\ensuremath{\mathsf{Grp}}\) and \(\ensuremath{\mathsf{Set}}\); see [Li1]. More explicitly, take a small category \(I\) and a functor \(\beta: I^{\operatorname{op}} \to \ensuremath{\mathsf{Grp}}\). As a set, \[\varprojlim U\beta = \left\{\begin{array}{r|l} (x_i)_i \in \prod_{i \in \operatorname{Ob}(I)} U\beta(i) & \forall [i \to j] \in \mathrm{Mor}(I), \\ & \beta(i \to j)(x_j) = x_i \end{array}\right\},\] and the projection \(p_i: \varprojlim U\beta \to U\beta(i)\) maps \((x_j)_j\) to \(x_i\). Our task is now to:

1. equip \(\varprojlim U\beta\) with a group structure such that every \(p_i\) is a group homomorphism, and show that this group structure is unique;

2. verify the universal property of \(\varprojlim \beta\) for this group together with the family of projection homomorphisms \((p_i)_i\).

Clearly, \(\varprojlim U\beta\) is a subgroup of the direct product \(\prod_i \beta(i)\). Regard this subgroup as \(\varprojlim \beta \in \operatorname{Ob}(\ensuremath{\mathsf{Grp}})\); this is the unique group structure making every \(p_i\) a group homomorphism. This completes the first task.

We next verify the universal property of the cone \(\left( \varprojlim \beta, (p_i)_i\right)\). Consider any cone \((L, (q_i)_i)\) in \(\ensuremath{\mathsf{Grp}}\), with \(q_i: L \to \beta(i)\). The universal property gives a unique map \(\varphi: UL \to \varprojlim U\beta\) such that \(p_i \varphi = q_i\) for every \(i\). Concretely, \(\varphi\) is in fact given by \(y \mapsto (q_i(y))_{i \in \operatorname{Ob}(I)}\), so this map is a group homomorphism \(L \to \varprojlim \beta\).² The verification is complete.

By an entirely similar argument, one can show that the forgetful functor from the category of compact Hausdorff spaces \(\ensuremath{\mathsf{CHaus}}\) to \(\ensuremath{\mathsf{Set}}\) also creates all small \(\varprojlim\). The corresponding verification is left as an exercise in this chapter. The next result concerns a functor category of the form \(\mathcal{C}^J\) (which may be a “large” category, but this causes no problem). We need two preparatory steps.

• Let \(J\) be a set. The product of \(J\) copies of \(\mathcal{C}\), namely \(\mathcal{C}^J\), is defined by \(\operatorname{Ob}\left(\mathcal{C}^J\right) = \operatorname{Ob}(\mathcal{C})^J\) and \(\mathrm{Mor}\left(\mathcal{C}^J \right) = \mathrm{Mor}(\mathcal{C})^J\). For every \(j \in J\) there is an evident projection functor \(p_j: \mathcal{C}^J \to \mathcal{C}\).

  Given a category \(I\) and a functor \(\alpha: I \to \mathcal{C}^J\), if \(\varinjlim (p_j \alpha)\) exists for every \(j\), then \(\varinjlim \alpha\) also exists. It is constructed by taking colimits “pointwise” or “objectwise”: \(\varinjlim \alpha = \left( \varinjlim (p_j \alpha) \right)_{j \in J}\).

• Let \(J\) be a category. We may consider the functor category \(\mathcal{C}^J\). For each \(j \in \operatorname{Ob}(J)\) there is an evaluation functor \(\mathrm{ev}_j: \mathcal{C}^J \to \mathcal{C}\), which maps an object \(F\) to \(Fj\) and a morphism \(\varphi = (\varphi_{j'})_{j' \in \operatorname{Ob}(J)}\) to \(\varphi_j\). We shall later consider functors of the form \(\alpha: I \to \mathcal{C}^J\) and their \(\varinjlim\). Observe that \(\alpha\) can be viewed as a functor \(I \times J \to \mathcal{C}\), whose value is denoted \(\alpha(i, j)\).

The two systems of notation are compatible. If \(J\) is a discrete category and is viewed as a set, then \(\mathcal{C}^J\) is simply the product discussed above. For a general \(J\), the forgetful functor \(\mathcal{F}: \mathcal{C}^J \to \mathcal{C}^{\operatorname{Ob}(J)}\) maps an object \(F\) to \((Fj)_{j \in \operatorname{Ob}(J)}\) and a morphism \(\varphi\) to \((\varphi_j)_{j \in \operatorname{Ob}(J)}\). Thus \(\mathrm{ev}_j = p_j \mathcal{F}\).

Proposition. Let \(J\) and \(\mathcal{C}\) be categories.

1. The forgetful functor \(\mathcal{F}: \mathcal{C}^J \to \mathcal{C}^{\operatorname{Ob}(J)}\) creates \(\varinjlim\) and \(\varprojlim\).

2. Let \(I\) be a category and suppose every functor \(I \to \mathcal{C}\) has a \(\varinjlim\) (or \(\varprojlim\)). Then all functors of the form \(I \to \mathcal{C}^J\) also have a \(\varinjlim\) (or \(\varprojlim\)) in \(\mathcal{C}^J\).

3. Under the assumption in (ii), for every \(j \in \operatorname{Ob}(J)\) the evaluation functor \(\mathrm{ev}_j: \mathcal{C}^J \to \mathcal{C}\) preserves \(\varinjlim\) (or \(\varprojlim\)) for diagrams of shape \(I\). In other words, limits in \(\mathcal{C}^J\) are likewise defined “pointwise” or “objectwise.”

Proof. It suffices to consider \(\varinjlim\). For (i), let \(\alpha: I \to \mathcal{C}^J\) and suppose \(\varinjlim \mathcal{F}\alpha\) exists and is given by the cocone \[\left( (X(j))_{j \in \operatorname{Ob}(J)}, \; \left( \iota_i = (\iota_{i, j})_{j \in \operatorname{Ob}(J)} \right)_{i \in \operatorname{Ob}(I)} \right)\] with \(\iota_{i,j}: \alpha(i, j) \to X(j)\). We wish to lift it to a cocone in \(\mathcal{C}^J\) that gives \(\varinjlim \alpha\).

For (ii), consider a functor \(\alpha: I \to \mathcal{C}^J\). By the preceding discussion, \(\mathcal{F}\alpha: I \to \mathcal{C}^{\operatorname{Ob}(J)}\) has a \(\varinjlim\), defined objectwise. By (i), \(\mathcal{F}\) creates \(\varinjlim \alpha\) in \(\mathcal{C}^J\),³ while the assertion in (iii) that \(\mathrm{ev}_j\) preserves this \(\varinjlim\) follows immediately from the construction.

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1. More precisely, \(\Delta\) here denotes the diagonal functor \(\mathcal{C} \to \mathcal{C}^I\); see Example 1.6.3. The notation \(\Delta\) was chosen from the initial letter of the pinyin word duìjiǎo (“diagonal”). By a pictorial analogy, however, one may just as well picture \(\Delta\) as a “cocone”; see diagram equation.↩︎

2. Translator’s note: the source writes \(L \to \varinjlim \beta\); the target corrects this to \(L \to \varprojlim \beta\), in accordance with the projective limit being proved (O014-C007).↩︎

3. Translator’s note: the source writes \(\mathcal{C}\) here. The target corrects this to \(\mathcal{C}^J\), the domain of the forgetful functor \(\mathcal{F}\) and the category containing the newly constructed functor \(X: J \to \mathcal{C}\); see correction O014-C009.↩︎

Filtered Inductive Limits

A familiar fact from mathematical analysis is that, when a sequence converges, its limit may be computed from any subsequence. In category theory, this idea is embodied by cofinality. We begin with connectedness.

Definition. Let \(I\) be a category. If \(\operatorname{Ob}(I) \neq \emptyset\) and, for every \(i, i' \in \operatorname{Ob}(I)\), there exist \(n \geq 1\) and morphisms \[i = i_0 \leftarrow i_1 \to i_2 \leftarrow i_3 \to i_4 \leftarrow \cdots \to i_{2n} = i' ,\] then \(I\) is called connected.

We introduce an auxiliary concept that will be used repeatedly in the next several sections.

Definition (Comma category; see [Li1).]

Given categories \(I\), \(J\), and a functor \(H: J \to I\), let \(i \in \operatorname{Ob}(I)\).

• By definition, the objects of the comma category \((i/H)\) (or \((H/i)\)) are data \((j, i \to Hj)\) (or \((j, Hj \to i)\)), where \(j \in \operatorname{Ob}(J)\) and \(i \to Hj\) (or \(Hj \to i\)) is a morphism in \(I\). A morphism from \((j, i \to Hj)\) to \((j', i \to Hj')\) (or from \((j, Hj \to i)\) to \((j', Hj' \to i)\)) is a morphism \(f: j \to j'\) in \(J\) making the following diagram commute: \[\text{Diagram chapter1-unit-014-d001} \quad \text{or} \quad \text{Diagram chapter1-unit-014-d002}\] Composition and identity morphisms are defined in the usual way.

• The two categories above come respectively with projection functors \(\Pi_{i/}: (i/H) \to J\) and \(\Pi_{/i}: (H/i) \to J\), taking an object \((j, i \to Hj)\) (or \((j, Hj \to i)\)) to \(j\) and a morphism \(f\) to \(f\).

• Every morphism \(i \to i'\) in \(I\) naturally induces functors \((i'/H) \to (i/H)\) and \((H/i) \to (H/i')\), making the following diagrams commute: \[\text{Diagram chapter1-unit-014-d003} \quad \text{Diagram chapter1-unit-014-d004}\]

Example. Regard a functor \(\alpha: I \to \mathcal{C}\) as an object of \(\mathcal{C}^I\), and consider the diagonal functor \(\Delta: \mathcal{C} \to \mathcal{C}^I\) taking every object to the corresponding constant functor. By Definition reference, \((\alpha/\Delta)\) is precisely the category of “cocones” introduced in Convention reference, while the functor \(\Pi_{\alpha/}\) maps each cocone to its vertex.

Definition. For a functor \(H: J \to I\), if \((i/H)\) is connected for every \(i \in \operatorname{Ob}(I)\), then \(H\) is called cofinal; if \(H\) is the inclusion functor of a subcategory, the subcategory \(J\) is called cofinal.

It is easy to verify that if \(I \to J\) and \(J \to K\) are both cofinal, then the composite functor \(I \to K\) is also cofinal. The verification is left to the reader as an exercise. The next result shows that a limit may be computed by restricting the diagram along a cofinal functor. Recall that \(H\) induces \(H^{\operatorname{op}}: J^{\operatorname{op}} \to I^{\operatorname{op}}\).

Proposition. If \(H: J \to I\) is cofinal, then:

• for a given functor \(\alpha: I \to \mathcal{C}\), \(\varinjlim \alpha\) exists if and only if \(\varinjlim \alpha H\) exists; in that case there is a canonical isomorphism \(\varinjlim \alpha H \simeq \varinjlim \alpha\);

• dually, for a given functor \(\beta: I^{\operatorname{op}} \to \mathcal{C}\), \(\varprojlim \beta\) exists if and only if \(\varprojlim \beta H^{\operatorname{op}}\) exists; in that case there is a canonical isomorphism \(\varprojlim \beta H^{\operatorname{op}} \simeq \varprojlim \beta\).

Proof. We discuss only \(\varinjlim\). In the language of §reference, it suffices to prove that the functor \((\alpha/\Delta) \to (\alpha H / \Delta)\) is an equivalence. It takes a cocone with base \(\alpha\), namely \((L, (f_i)_i)\), to the cocone with base \(\alpha H\), namely \((L, (f_{Hj})_j)\).

We next show that the functor is full and faithful. Giving a morphism in \((\alpha/\Delta)\), namely \((L, (f_i)_i) \to (L', (f'_i)_i)\), is equivalent to giving a morphism \(\theta: L \to L'\) in \(\mathcal{C}\) such that \(\theta f_i = f'_i\) for every \(i\). It suffices to check this condition on the objects \(H(j)\) in the cofinal image of \(H\): for every \(i\), choose \(j \in \operatorname{Ob}(J)\) and \(i \to H(j)\); the condition then becomes \(\theta f_{H(j)} = f'_{H(j)}\), which says precisely that \(\theta\) is a morphism in \((\alpha H / \Delta)\).

Definition.

Following [Li1], a category \(I\) with \(\operatorname{Ob}(I)\neq\emptyset\) satisfying the following conditions is called filtered¹.

• For every \(i, j \in \operatorname{Ob}(I)\), there is an object \(k \in \operatorname{Ob}(I)\) together with morphisms \(i \rightarrow k\) and \(j \rightarrow k\).

• For every pair of morphisms \(f, g: i \to j\) in \(I\), there is an object \(k \in \operatorname{Ob}(I)\) and a morphism \(h: j \to k\) such that \(hf = hg\).

If \(I\) is a filtered category and \(\alpha: I \to \mathcal{C}\) is a functor, the corresponding \(\varinjlim \alpha\) is called a filtered inductive limit.

Example.

One common class of filtered categories comes from filtered partially ordered sets². Every partially ordered set \((P, \leq)\) determines a corresponding category \(\mathcal{P}\) such that \(x \leq y \iff \operatorname{Hom}_{\mathcal{P}}(x, y) \neq \emptyset\). If \(\mathcal{P}\) is a filtered category, then \((P, \leq)\) is called a filtered partially ordered set. A partially ordered set is filtered if and only if it is nonempty and every pair of elements \(i, j\) has a common upper bound \(k\). Every nonempty totally ordered set is plainly filtered.

Proposition. Let \(I\) be a filtered category. A full subcategory \(J\subset I\) is cofinal if and only if, for every \(i \in \operatorname{Ob}(I)\), there are \(j \in \operatorname{Ob}(J)\) and a morphism \(i \to j\); in that case \(J\) is also filtered.

To prove that \(J\) is filtered, take arbitrary \(i, j \in \operatorname{Ob}(J)\). Since \(I\) is filtered, there are \(k_0 \in \operatorname{Ob}(I)\) and morphisms \(i\to k_0\) and \(j\to k_0\); also choose \(k \in \operatorname{Ob}(J)\) and a morphism \(k_0 \to k\). Composing these morphisms yields morphisms \(i \rightarrow k \leftarrow j\) in \(J\).³ This proves the first condition for a filtered category. The second is proved by the same idea.

Example. Define \(\mathrm{OFin}_I := \left\{ \text{full subcategories}\; J \subset I: \operatorname{Ob}(J) \;\text{is finite} \right\}\). Ordered by \(\subset\), this is a filtered partially ordered set: for any \(J, K \in \mathrm{OFin}_I\), the subcategory obtained by taking the union of their sets of objects is a common upper bound for \(J\) and \(K\).

To see the usefulness of this construction, consider an arbitrary functor \(\alpha: I \to \mathcal{C}\) (or \(\beta: I^{\operatorname{op}} \to \mathcal{C}\)). Its restriction to a full subcategory \(J\) gives \(\alpha|_J\) (or \(\beta|_{J^{\operatorname{op}}}\)). If \(J \subset K\), there is a natural morphism \(\varinjlim \alpha|_J \to \varinjlim \alpha|_K\) (or \(\varprojlim \beta|_{K^{\operatorname{op}}} \to \varprojlim \beta|_{J^{\operatorname{op}}}\)), provided these limits exist. The next result explains how an arbitrary limit can be approximated “in a filtered manner” by limits over subcategories having only finitely many objects.

Proposition. In the situation above, there is a canonical isomorphism \[\varinjlim_{\substack{J \in \mathrm{OFin}_I \\ \subset}} \varinjlim \alpha|_J \ensuremath{\stackrel{\sim}{\rightarrow}}\varinjlim \alpha\] (or \(\varprojlim \beta \ensuremath{\stackrel{\sim}{\rightarrow}} \varprojlim_{J \in \mathrm{OFin}_I^{\operatorname{op}}} \varprojlim \beta|_{J^{\operatorname{op}}}\)), provided that the iterated limit on the left (or on the right) exists.⁴

For every \(J\), the concrete construction of the \(\varprojlim\) of sets is \[\varprojlim_{j \in \operatorname{Ob}(J)} X_j = \left\{ (x_j)_j \in \prod_{j \in \operatorname{Ob}(J)} X_j : \forall [i \to j] \in \operatorname{Mor}(J), \; f_{i \to j}(x_j) = x_i \right\},\] so the bijection is clear: every piece of data defining \(\varprojlim_i X_i\), whether a coordinate \(x_i\) or a compatibility condition \(f_{i \to j}(x_j) = x_i\), is contained in some \(J \in \mathrm{OFin}_I\).

Recall that the category \(\ensuremath{\mathsf{Set}}\) has all small \(\varinjlim\) and \(\varprojlim\). Small filtered inductive limits have a particularly simple description.

Proposition. Let \(I\) be a small filtered category and \(\alpha: I \to \ensuremath{\mathsf{Set}}\) a functor. Define the following binary relation on the set \(\bigsqcup_{i \in \operatorname{Ob}(I)} \alpha(i)\): for \(x \in \alpha(i)\) and \(y \in \alpha(j)\), the relation \(x \sim y\) means that there is an object \(k\in\operatorname{Ob}(I)\) together with morphisms \(i\rightarrow k\) and \(j\rightarrow k\) such that \(\alpha(i \to k)(x) = \alpha(j \to k)(y)\). Then \(\sim\) is an equivalence relation, and \[\varinjlim \alpha = \left( \bigsqcup_{i \in \operatorname{Ob}(I)} \alpha(i) \right) \big/ \sim .\]

Proof. See the discussion following [Li1].

An element of the preceding \(\varinjlim \alpha\) will henceforth be written \([x_{i_0}]\), where \(i_0 \in \operatorname{Ob}(I)\) and \([x_{i_0}]\) is the equivalence class containing \(x_{i_0} \in \alpha(i_0)\). On the other hand, for a general small category \(J\) and functor \(\beta: J^{\operatorname{op}} \to \ensuremath{\mathsf{Set}}\), we write an element of \(\varprojlim \beta\) as \((y_j)_{j \in \operatorname{Ob}(J)}\), where \(y_j \in \beta(j)\) satisfies the compatibility condition \(\beta(j' \to j)(y_j) = y_{j'}\). Next consider a functor of the form \(\alpha: I \times J^{\operatorname{op}} \to \ensuremath{\mathsf{Set}}\), with \(I, J\) small categories; one may take \(\varinjlim\) and \(\varprojlim\) in the variables \(I\) and \(J\), respectively. For every \((i_0, j_0) \in \operatorname{Ob}(I \times J)\), there is a composite of natural maps \[\varprojlim_j \alpha(i_0, j) \to \alpha(i_0, j_0) \to \varinjlim_i \alpha(i, j_0);\] this composite is natural in \(i_0\) and \(j_0\). Varying \(j_0\) gives a canonical map \(\varprojlim_j \alpha(i_0, j) \to \varprojlim_j \varinjlim_i \alpha(i, j)\); then varying \(i_0\) gives a canonical map \[\begin{equation} e: \varinjlim_i \varprojlim_j \alpha(i, j) \to \varprojlim_j \varinjlim_i \alpha(i, j). \end{equation}\] In terms of representatives, \(e\) maps \(\left[ (x_{i_0, j})_j \right]\) to \(\left( [ x_{i_0, j}] \right)_j\). If \(\operatorname{Mor}(J)\) is a finite set, the category \(J\) is called finite; equivalently, its object set is finite and every morphism set between two objects is finite. The next result will be used in §1.9.

Proposition. Let \(I\) be a small filtered category, \(J\) a finite category, and \(\alpha: I \times J^{\operatorname{op}} \to \ensuremath{\mathsf{Set}}\) a functor. Then the map \(e\) in equation is a bijection.

Proof. We first show that \(e\) is surjective. Given \((b_j)_{j \in \operatorname{Ob}(J)} \in \varprojlim_j \varinjlim_i \alpha(i, j)\), write each \(b_j\) as \([x_{i_j, j}]\). Since \(I\) is filtered and \(J\) is finite, we may choose a common target object \(i\in\operatorname{Ob}(I)\) for all \(i_j\), and replace each representative by its image along a morphism \(i_j\to i\). Thus all representatives may be written \(x_{i,j}\) with the same index \(i\). For every morphism \(j \to j'\) in \(J\), we have \[\left[ \alpha(i, j \to j') (x_{i, j'}) \right] = \left[ x_{i, j} \right].\] Since \(\operatorname{Mor}(J)\) is finite and \(I\) is filtered, we may choose one morphism \(i\to i_1\) that equalizes all these pairs, replace all representatives by their images at index \(i_1\), and then rename \(i_1\) as \(i\). Thus \(\alpha(i, j \to j') (x_{i, j'}) = x_{i, j}\) holds for every \([j \to j'] \in \operatorname{Mor}(J)\). Consequently, \(e\left( [(x_{i, j})_j] \right) = (b_j)_j\).

We next show that \(e\) is injective. Suppose \(e(x) = e(y)\). Since \(I\) is filtered, we may replace representatives of \(x\) and \(y\) by their images at one common index \(i\in\operatorname{Ob}(I)\), so that they have representatives \((x_{i,j})_j\) and \((y_{i,j})_j\). Thus \([x_{i,j}]=[y_{i,j}]\) for every \(j \in \operatorname{Ob}(J)\). Since \(I\) is filtered and \(J\) is finite, we may choose one common later index, replace all components by their images there, and rename it \(i\), so that \(x_{i,j}=y_{i,j}\) for every \(j\). Hence \(x=y\).

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1. Translator’s note: the source does not state the condition \(\operatorname{Ob}(I)\neq\emptyset\), so the empty category would satisfy both items vacuously; in the second item, the source also calls \(k\in\operatorname{Ob}(I)\) a morphism. The target adds the nonemptiness condition and corrects the type of \(k\) to an object (O014-C010 and O014-C011).↩︎

2. When used to index limits, a filtered category can often be replaced by a filtered partially ordered set; see Remark A.2.3 and the explanations for the exercises in Appendix A.↩︎

3. Translator’s note: the source calls \(i\rightarrow k_0\leftarrow j\) morphisms in \(J\), although \(k_0\) is only known to lie in \(I\). The target uses the composites into \(k\in\operatorname{Ob}(J)\) (O014-C012).↩︎

4. Translator’s note: in the dual formula, the source writes an outer inductive limit. For \(J\subset K\), however, the transition morphism goes from the \(K\)-term to the \(J\)-term, so the terms form a diagram on \(\mathrm{OFin}_I^{\operatorname{op}}\) and the outer construction is a projective limit (O014-C013).↩︎

Kan Extensions

Definition (D. Kan).

Consider categories \(\mathcal{C}\), \(\mathcal{D}\), \(\mathcal{E}\) and functors \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\).

• A left Kan extension of the functor \(F\) along \(K\) is the following data \((\operatorname{Lan}_K F, \eta)\):

  • \(\operatorname{Lan}_K F: \mathcal{D} \to \mathcal{E}\) is a functor;

  • \(\eta: F \to (\operatorname{Lan}_K F) K\) is a natural transformation.

  These data must satisfy the following universal property: for every \(L: \mathcal{D} \to \mathcal{E}\) and \(\xi: F \to LK\), there is a unique natural transformation \(\chi: \operatorname{Lan}_K F \to L\) such that \(\xi = (\chi K) \eta\) (using vertical and horizontal composition of natural transformations), or, in \(2\)-cell diagrams: \[\text{Diagram chapter1-unit-015-d002} = \text{Diagram chapter1-unit-015-d003}.\]

• A right Kan extension of the functor \(F\) along \(K\) is the following data \((\operatorname{Ran}_K F, \varepsilon)\):

  • \(\operatorname{Ran}_K F: \mathcal{D} \to \mathcal{E}\) is a functor;

  • \(\varepsilon: (\operatorname{Ran}_K F)K \to F\) is a natural transformation.

  These data must satisfy the following universal property: for every \(R: \mathcal{D} \to \mathcal{E}\) and \(\delta: RK \to F\), there is a unique natural transformation \(\theta: R \to \operatorname{Ran}_K F\) such that \(\delta = \varepsilon (\theta K)\), or, in \(2\)-cell diagrams: \[\text{Diagram chapter1-unit-015-d004} = \text{Diagram chapter1-unit-015-d005}.\]

If \(\mathcal{C}\), \(\mathcal{D}\), and \(\mathcal{E}\) in the definition are replaced respectively by \(\mathcal{C}^{\operatorname{op}}\), \(\mathcal{D}^{\operatorname{op}}\), and \(\mathcal{E}^{\operatorname{op}}\), the pattern of directions of the functors remains the same, while the natural transformations reverse direction. Thus left and right Kan extensions are dual notions. Using the functor categories \(\mathcal{E}^{\mathcal{C}}\) and \(\mathcal{E}^{\mathcal{D}}\), the universal properties of left and right Kan extensions state, respectively, the bijections \[\begin{equation} \begin{aligned} \operatorname{Hom}_{\mathcal{E}^{\mathcal{D}}}\left( \operatorname{Lan}_K F, L \right) & \xrightarrow{1:1} \operatorname{Hom}_{\mathcal{E}^{\mathcal{C}}}\left( F, LK \right) \\ \chi & \longmapsto (\chi K)\eta , \\ \operatorname{Hom}_{\mathcal{E}^{\mathcal{D}}}\left( R, \operatorname{Ran}_K F \right) & \xrightarrow{1:1} \operatorname{Hom}_{\mathcal{E}^{\mathcal{C}}}\left( RK, F \right) \\ \theta & \longmapsto \varepsilon (\theta K); \end{aligned} \end{equation}\] their inverses are the maps \((L, \xi) \mapsto \chi\) and \((R, \delta) \mapsto \theta\), respectively, from Definition reference.

Proposition. Given categories \(\mathcal{C}\), \(\mathcal{D}\), \(\mathcal{E}\) and functors \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\), consider the pullback (that is, precomposition) functor \(K^*: \mathcal{E}^{\mathcal{D}} \to \mathcal{E}^{\mathcal{C}}\) between the functor categories.

1. If it exists, a left Kan extension \((\operatorname{Lan}_K F, \eta)\) is unique up to a unique isomorphism in \(\mathcal{E}^{\mathcal{D}}\); the same holds for a right Kan extension.

2. If the left Kan extension (or right Kan extension) along \(K\) exists for every \(F\), these extensions assemble into a left adjoint to \(K^*\), \(\operatorname{Lan}_K: \mathcal{E}^{\mathcal{C}} \to \mathcal{E}^{\mathcal{D}}\) (or a right adjoint \(\operatorname{Ran}_K: \mathcal{E}^{\mathcal{C}} \to \mathcal{E}^{\mathcal{D}}\)); the family of natural transformations assembled from \(\eta\) (or \(\varepsilon\)) in Definition reference is exactly the unit (or counit) of this adjunction.

Proof. For (i), simply apply the universal property in Definition reference in the usual way.

For (ii), observe that \(LK = K^*(L)\) and \(RK = K^*(R)\); the desired adjunction is precisely the content of equation. The assertion about the unit or counit follows similarly; see [Li1].

By convention, the data \(\eta\) or \(\varepsilon\) in a Kan extension are often omitted from the notation. If the left Kan extension (or right Kan extension) along \(K\) exists for every \(F\), the functor \(\operatorname{Lan}_K\) (or \(\operatorname{Ran}_K\)) obtained in (ii) also has the corresponding uniqueness property, by uniqueness of adjoints [Li1].

Remark. Since the definition of a Kan extension involves only composition of \(2\)-cells, it extends to general \(2\)-categories; the present situation is the special case of the \(2\)-category \(\ensuremath{\mathsf{Cat}}\). See [Li1] for details.

For every category \(\mathcal{C}\), denote the unique functor \(\mathcal{C} \to \mathbf{1}\) by \(!\); specifying a functor \(\mathbf{1} \to \mathcal{C}\) is equivalent to specifying an object of \(\mathcal{C}\). For every category \(\mathcal{E}\) and object \(X\) in it, write \(\Delta(X): \mathcal{C} \to \mathcal{E}\) for the composite \(\mathcal{C} \xrightarrow{!} \mathbf{1} \xrightarrow{\text{constant}\; X} \mathcal{E}\); this is the constant functor that sends every object to \(X\) and every morphism to \(\ensuremath{\mathrm{id}}_X\).

For example, consider \((\operatorname{Lan}_! F, \eta)\). The functor \(\operatorname{Lan}_! F: \mathbf{1} \to \mathcal{E}\) may be viewed as an object of \(\mathcal{E}\), while \(\eta\) is a natural transformation \(F \to \Delta(\operatorname{Lan}_! F)\) in \(\mathcal{E}^{\mathcal{C}}\). Its universal property is equivalent to the assertion that the data \((\operatorname{Lan}_! F, \eta)\) give an initial object of the category \((F/\Delta)\); this is precisely the description of \(\varinjlim F\) in §reference.

This adjunction gives a canonical bijection \(\operatorname{Hom}_{\mathcal{C}^{\mathcal{D}}}(\underbracket{G^* \ensuremath{\mathrm{id}}_{\mathcal{C}}}_{= G}, L) \ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{Hom}_{\mathcal{C}^{\mathcal{C}}}(\ensuremath{\mathrm{id}}_{\mathcal{C}} , \underbracket{F^* L}_{=LF})\), for arbitrary \(L: \mathcal{D} \to \mathcal{C}\); when \(L = G\), \(\ensuremath{\mathrm{id}}_G\) maps to \(\eta\). Writing the universal property of \(\operatorname{Lan}_F (\ensuremath{\mathrm{id}}_{\mathcal{C}})\) as the bijection in equation, one sees at once that \((G, \eta)\) gives \(\operatorname{Lan}_F (\ensuremath{\mathrm{id}}_{\mathcal{C}})\). The argument for \(\operatorname{Ran}_G (\ensuremath{\mathrm{id}}_{\mathcal{D}})\) is entirely similar.¹

In practice, some situations require a further strengthening of the notion of Kan extension², especially in the later treatment of derived functors.

Definition. Consider categories \(\mathcal{C}\), \(\mathcal{D}\), \(\mathcal{E}\), \(\mathcal{F}\) and functors \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\).³

• Suppose the right Kan extension \((\operatorname{Ran}_K F, \varepsilon)\) exists. A functor \(M: \mathcal{E} \to \mathcal{F}\) is said to preserve \(\operatorname{Ran}_K F\) if the composite \(2\)-cell \[\text{Diagram chapter1-unit-015-d009} =: \text{Diagram chapter1-unit-015-d010}\] gives a right Kan extension of \(MF\) along \(K\).

• Suppose the left Kan extension \((\operatorname{Lan}_K F, \eta)\) exists. A functor \(M: \mathcal{E} \to \mathcal{F}\) is said to preserve \(\operatorname{Lan}_K F\) if the composite \(2\)-cell \[\text{Diagram chapter1-unit-015-d011} =: \text{Diagram chapter1-unit-015-d012}\] gives a left Kan extension of \(MF\) along \(K\).

• If a left Kan extension \((\operatorname{Lan}_K F, \eta)\) (or a right Kan extension \((\operatorname{Ran}_K F, \varepsilon)\)) is preserved by every functor whose domain is \(\mathcal{E}\), the Kan extension is called absolute.

Proof.

The case of \(\underline{\eta}\) is dual. It suffices to observe that \((\mathrm{R}G)\alpha: \mathrm{R}G \circ \mathrm{L}F \circ K \to \mathrm{R}G \circ K' \circ F\) also makes \(\mathrm{R}G \circ \mathrm{L}F\) a right Kan extension, because \(\mathrm{L}F\) is an absolute right Kan extension.

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1. Translator’s note: the source writes \(\operatorname{Ran}_F(\ensuremath{\mathrm{id}}_{\mathcal{C}})\) in the final sentence, contrary to the preceding assertion concerning \((F,\varepsilon)\). The target corrects this to \(\operatorname{Ran}_G(\ensuremath{\mathrm{id}}_{\mathcal{D}})\) (O014-C014).↩︎

2. One common strengthening is called a pointwise Kan extension; it is not discussed here.↩︎

3. Translator’s note: the source does not include the category \(\mathcal{F}\) in the initial list, even though it is used as the codomain of \(M\). The target adds it (O014-C015).↩︎

Constructing Kan Extensions by Limits

As in §reference, throughout this section we continue to consider functors \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\), with the aim of studying the existence of \(\operatorname{Lan}_K F\) and \(\operatorname{Ran}_K F\). Recall the discussion at the beginning of §reference. The motivation for Kan extensions is to seek a functor \(G: \mathcal{D} \to \mathcal{E}\) such that \(F \simeq GK\), or at least to find the best approximation to one. How should \(Gd\) be defined for \(d \in \operatorname{Ob}(\mathcal{D})\)? The only clue is that when \(d=Kc\), the object \(Gd\) ought to be \(Fc\), up to isomorphism. This suggests approximating \(Gd\) by a colimit \(\varinjlim\) (or a limit \(\varprojlim\)) of the objects \(Fc\), indexed by all \(c \in \operatorname{Ob}(\mathcal{C})\) and morphisms \(Kc \to d\) (or \(d \to Kc\)). These two constructions have dual universal properties. We now make the construction precise. Given \(d \in \operatorname{Ob}(\mathcal{D})\), define the categories \((K/d)\) and \((d/K)\) as in Definition reference, together with the projection functors \[\Pi_{/d}: (K/d) \to \mathcal{C}, \quad \Pi_{d/}: (d/K) \to \mathcal{C}.\] Our focus will be on the composite functors \(F\Pi_{/d}: (K/d) \to \mathcal{E}\) and \(F\Pi_{d/}: (d/K) \to \mathcal{E}\), which send the objects \((c,Kc \to d)\) and \((c,d \to Kc)\), respectively, to \(Fc\). Assuming that the limits under discussion exist, we first record a few observations.

• Every morphism \(d \to d'\) induces a canonical morphism \(\varinjlim F\Pi_{/d} \to \varinjlim F\Pi_{/d'}\); it is characterized by requiring the diagram \[\begin{equation} \text{Diagram chapter1-unit-016-d001} \end{equation}\] to commute for every \((c,Kc \to d) \in \operatorname{Ob}((K/d))\) (which induces an object \((c,Kc \to d') \in \operatorname{Ob}((K/d'))\)), where the vertical arrows are the structure morphisms of the colimits. This is an instance of the functoriality of colimits; see [Li1]¹ and its proof.

• Considering \((c,Kc \xrightarrow{\ensuremath{\mathrm{id}}} Kc) \in \operatorname{Ob}(K/Kc)\) gives a canonical morphism \(\eta_c: Fc \to \varinjlim F\Pi_{/Kc}\).

• Every morphism \(c \to c'\) induces a canonical morphism \(\varinjlim F\Pi_{/Kc} \to \varinjlim F\Pi_{/Kc'}\) that makes the following diagram commute: \[\text{Diagram chapter1-unit-016-d002}\]

By duality, \(\varprojlim F\Pi_{d/}\) has the corresponding properties, including a canonical morphism \(\varepsilon_c: \varprojlim F\Pi_{Kc/} \to Fc\). The next theorem is based on this functoriality.

Theorem. Consider functors \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\).

1. Suppose that for every \(d \in \operatorname{Ob}(\mathcal{D})\), the colimit \(\varinjlim(F\Pi_{/d})\) exists in \(\mathcal{E}\). Then \((\operatorname{Lan}_K F)(d) := \varinjlim(F\Pi_{/d})\), together with the functoriality given by equation, determines the left Kan extension \(\operatorname{Lan}_K F: \mathcal{D} \to \mathcal{E}\). The associated natural transformation \(\eta: F \to (\operatorname{Lan}_K F)\circ K\) comes from the family \((\eta_c)_{c\in\operatorname{Ob}(\mathcal{C})}\) discussed above.

2. Suppose that for every \(d \in \operatorname{Ob}(\mathcal{D})\), the limit \(\varprojlim(F\Pi_{d/})\) exists in \(\mathcal{E}\). Then \((\operatorname{Ran}_K F)(d) := \varprojlim(F\Pi_{d/})\), together with the dual of equation, determines the right Kan extension \(\operatorname{Ran}_K F: \mathcal{D} \to \mathcal{E}\). The associated natural transformation \(\varepsilon: (\operatorname{Ran}_K F)\circ K \to F\) comes from the family \((\varepsilon_c)_{c\in\operatorname{Ob}(\mathcal{C})}\) discussed above.

3. If \(K: \mathcal{C} \to \mathcal{D}\) is fully faithful, then the \(\eta\) constructed in (i) (or the \(\varepsilon\) constructed in (ii)) is an isomorphism.

Remark. If \(\mathcal{C}\) is a small category, then \((K/d)\) and \((d/K)\) are also small categories for every \(d\in\operatorname{Ob}(\mathcal{D})\). Consequently, when \(\mathcal{E}\) is cocomplete (or complete), Theorem reference and Proposition reference show that \(K^*: \mathcal{E}^{\mathcal{D}} \to \mathcal{E}^{\mathcal{C}}\) has a left adjoint \(\operatorname{Lan}_K\) (or a right adjoint \(\operatorname{Ran}_K\)).

Example (Inverse image of presheaves).

Take \(\mathcal{E}:=\ensuremath{\mathsf{Set}}\), which is both complete and cocomplete. Let \(f:X\to Y\) be a continuous map of topological spaces. Let \(\mathcal{C}:=\ensuremath{\mathsf{Open}}_Y\) be the category whose objects are the open subsets of \(Y\) and whose morphisms are inclusions of open subsets; similarly, let \(\mathcal{D}:=\ensuremath{\mathsf{Open}}_X\). Now define a functor \(\ensuremath{\mathsf{Open}}_Y\to\ensuremath{\mathsf{Open}}_X\) that sends an open subset \(V\subset Y\) to \(f^{-1}V\), and regard it below as a functor \(K:\ensuremath{\mathsf{Open}}_Y^{\operatorname{op}}\to\ensuremath{\mathsf{Open}}_X^{\operatorname{op}}\). Readers familiar with sheaf theory will recognize that the objects of \(\ensuremath{\mathsf{Open}}_Y^\wedge:=\ensuremath{\mathsf{Set}}^{\ensuremath{\mathsf{Open}}_Y^{\operatorname{op}}}\) are, by definition, precisely the presheaves on \(Y\), while \(K^*:\ensuremath{\mathsf{Open}}_X^\wedge\to\ensuremath{\mathsf{Open}}_Y^\wedge\) is the direct-image functor between the presheaf categories: it sends a presheaf \(\mathcal{F}\) on \(X\) to the presheaf on \(Y\) given by \(V\mapsto\mathcal{F}(f^{-1}V)\). In this situation, the left adjoint of the direct image is \(\operatorname{Lan}_K\), given by Theorem reference; it sends a presheaf \(\mathcal{G}\) on \(Y\) to \[\begin{gather*} U \mapsto \varinjlim_{\substack{V \subset Y: \text{open subset} \\ \text{such that}\; U \subset f^{-1} V }} \mathcal{G}(V), \quad U \subset X: \text{open subset}, \\ U \subset f^{-1}V \;\xleftrightarrow{\text{corresponds to}}\; \text{a morphism}\; K(V) \to U \;\text{in}\; \ensuremath{\mathsf{Open}}_X^{\operatorname{op}}. \end{gather*}\] Unsurprisingly, this is precisely the inverse image of \(\mathcal{G}\) in sheaf theory.

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1. Translator’s note: the source gives the locator as Lemma 2.7,4; it has been corrected to Lemma 2.7.4 here (O014-C016).↩︎

Gabriel–Zisman Localization

Gabriel–Zisman localization, henceforth simply called localization, is the most economical way to adjoin inverses to a family of morphisms in a category. The theory originates in [GZ67] and is characterized by a universal property.

Definition (P. Gabriel, M. Zisman).

Let \(\mathcal{C}\) be a category and let \(S\) be a subset of \(\operatorname{Mor}(\mathcal{C})\) that contains all identity morphisms and is closed under composition. A localization of \(\mathcal{C}\) at \(S\) means a category \(\mathcal{C}[S^{-1}]\) (which may be a “large” category), together with a functor \(Q:\mathcal{C}\to\mathcal{C}[S^{-1}]\), called the localization functor, satisfying the following conditions.

• For every \(s\in S\), its image \(Q(s)\) is an isomorphism in \(\mathcal{C}[S^{-1}]\).

• For every category \(\mathcal{D}\) (again possibly “large”) and functor \(F:\mathcal{C}\to\mathcal{D}\), if \(F\) sends every morphism in \(S\) to an isomorphism, then there is a unique functor \(F[S^{-1}]:\mathcal{C}[S^{-1}]\to\mathcal{D}\) such that \(F=F[S^{-1}]Q\).

By convention, we often omit \(Q\) from the data. Some sources, such as [KS06], give a somewhat weaker universal property for localization¹. We next show that localization is unique up to a unique isomorphism.

Proof. Since \(Q'\) sends \(S\) to isomorphisms, the universal property gives a unique \(G\) such that \(GQ=Q'\). Similarly, there is a unique \(G'\) such that \(G'Q'=Q\). Since \(G'GQ=Q\), taking \(\mathcal{D}=\mathcal{C}[S^{-1}]\) and \(F=Q\) in the universal property gives \(G'G=\ensuremath{\mathrm{id}}_{\mathcal{C}[S^{-1}]}\). Likewise, \(GG'=\ensuremath{\mathrm{id}}_{\mathcal{C}[S^{-1}]'}\)².

Proposition. Suppose that the localization \((\mathcal{C}[S^{-1}],Q)\) exists. Let \(S^{\operatorname{op}}\) denote the corresponding image of \(S\) in \(\operatorname{Mor}(\mathcal{C}^{\operatorname{op}})\). Then there is an equivalence of categories, compatible with \(Q\), \(\mathcal{C}[S^{-1}]^{\operatorname{op}}\leftrightarrows \mathcal{C}^{\operatorname{op}}[(S^{\operatorname{op}})^{-1}]\).

Proof. By the uniqueness of localization, it suffices to apply \((-)^{\operatorname{op}}\) to all the categories and functors in Definition reference, since this operation does not reverse the direction of a functor.

Proposition. The localization functor \(Q:\mathcal{C}\to\mathcal{C}[S^{-1}]\) is essentially surjective. Moreover, for every category \(\mathcal{D}\) and functors \(A,B:\mathcal{C}[S^{-1}]\to\mathcal{D}\), the evident map \[\operatorname{Hom}_{\mathcal{D}^{\mathcal{C}[S^{-1}]}}(A,B) \to \operatorname{Hom}_{\mathcal{D}^{\mathcal{C}}}\left(AQ,BQ\right)\] is a bijection. In other words, \(Q^*:\mathcal{D}^{\mathcal{C}[S^{-1}]}\to\mathcal{D}^{\mathcal{C}}\) is a fully faithful functor.

Proof. We may realize the localization concretely as above. Write the map induced by \(Q\) on objects as \(X\mapsto\underline{X}\). It is a bijection, hence \(Q\) is essentially surjective. It follows at once that the map \(\operatorname{Hom}_{\mathcal{D}^{\mathcal{C}[S^{-1}]}}(A,B)\to \operatorname{Hom}_{\mathcal{D}^{\mathcal{C}}}(AQ,BQ)\) is injective.

Definition.

A subset \(S\subset\operatorname{Mor}(\mathcal{C})\) satisfying the following properties is called a left multiplicative system in \(\mathcal{C}\).

1. For every \(X\in\operatorname{Ob}(\mathcal{C})\), one has \(\ensuremath{\mathrm{id}}_X\in S\).

2. For all \(f,g\in S\), if the composite \(gf\) is defined, then \(gf\in S\).

3. Given morphisms \(X\xrightarrow{s\in S}Z\xleftarrow{f}Y\), there are morphisms in \(\mathcal{C}\), \(X\xleftarrow{f'}W\xrightarrow{s'\in S}Y\), such that \(sf'=fs'\), depicted by the commutative diagram \[\text{Diagram chapter1-unit-017-diagram-004} \quad \text{commutative}.\]

4. Let \(f,g:X\to Y\) be morphisms in \(\mathcal{C}\) and let \(s:Y\to W\) belong to \(S\). If \(sf=sg\), then there is a morphism \(t:Z\to X\) in \(S\) such that \(ft=gt\). Diagrammatically, \[\text{Diagram chapter1-unit-017-d003}\]

The dashed parts of these diagrams are the morphisms whose existence is asserted.

If the image \(S^{\operatorname{op}}\) of \(S\) in \(\operatorname{Mor}(\mathcal{C}^{\operatorname{op}})\) is a left multiplicative system, then \(S\) is called a right multiplicative system in \(\mathcal{C}\). Equivalently, reverse all arrows in conditions (S3) and (S4). A subset \(S\) that is both a left and a right multiplicative system is called a multiplicative system in \(\mathcal{C}\).

Convention. For a given left or right multiplicative system \(S\), from now on arrows belonging to \(S\) will always be drawn as \(\rightarrowtail\) in commutative diagrams, to distinguish them from the other arrows.

Fix \(X\in\operatorname{Ob}(\mathcal{C})\). In accordance with this convention, define the category \(S_{/X}\) (or \(S_{X/}\)) for a left (or right) multiplicative system \(S\) as follows.

Lemma. If \(S\) is a left (or right) multiplicative system, then \(S_{/X}^{\operatorname{op}}\) (or \(S_{X/}\)) is a filtered category.

Let \(X,Y\in\operatorname{Ob}(\mathcal{C})\). We now define sets \(M_{X,Y}^l\) and \(M_{X,Y}^r\) for left and right multiplicative systems, respectively. Note that they need not be small sets.

• Let \(S\) be a left multiplicative system. Abbreviate a diagram in \(\mathcal{C}\) of the form \[\text{Diagram chapter1-unit-017-diagram-011}\] by \((Z;s,a)\). These data form the set \(M_{X,Y}^l\).

• Let \(S\) be a right multiplicative system. Abbreviate a diagram in \(\mathcal{C}\) of the form \[\text{Diagram chapter1-unit-017-diagram-012}\] by \((Z;a,s)\). These data form the set \(M_{X,Y}^r\).

Proof. Consider the functor \(\alpha:S_{/X}^{\operatorname{op}}\to\ensuremath{\mathsf{Set}}\) sending an object \(X\leftarrowtail Z\) to \(\operatorname{Hom}(Z,Y)\). By definition, \[M_{X, Y}^l = \bigsqcup_{X \leftarrowtail Z} \alpha(X \leftarrowtail Z),\] and since \(S_{/X}^{\operatorname{op}}\) is filtered (Lemma reference), it is easy to check that the binary relation \(\sim\) on \(M_{X,Y}^l\) arising from equation translates into the equivalence relation in Proposition reference. This also proves the bijection with \(\varinjlim\). The argument for the \(M_{X,Y}^r\) version is identical.

When discussing filtered colimits of sets, §reference assumes that the indexing category is small, so that the resulting colimit still lies in \(\ensuremath{\mathsf{Set}}\). Here \(S_{/X}\) and \(S_{Y/}\) need not be small categories. The assertion itself, however, does not depend on set size, and the argument can be formulated so as not to depend on a choice of Grothendieck universe.

The equivalence class \([Z;s,a]\) (or \([Z;a,s]\)) should be understood as the morphism \(as^{-1}\) (or \(s^{-1}a\)) in the category \(\mathcal{C}[S^{-1}]\) to be constructed.

The case in which \((U;s,a)\) is fixed and \((V;t,b)\) varies within its equivalence class is similar.

Definition–Theorem. Let \(S\) be a left multiplicative system in a category \(\mathcal{C}\).

1. There is a category \(\mathcal{C}[S^{-1}]^l\) (possibly a large category) whose object set is \(\operatorname{Ob}(\mathcal{C})\) and whose \(\operatorname{Hom}\) set between two objects \(X,Y\) is \(M_{X,Y}^l/\sim\). The identity morphism of \(X\) is \([X;\ensuremath{\mathrm{id}}_X,\ensuremath{\mathrm{id}}_X]\), and composition is given by the binary operation in Definition–Proposition reference.

2. There is a functor \(Q^l:\mathcal{C}\to\mathcal{C}[S^{-1}]^l\) that is the identity on objects and sends a morphism \(f:X\to Y\) to \([X;\ensuremath{\mathrm{id}}_X,f]\).

The dual statement holds for a right multiplicative system \(S\); the corresponding data are denoted \(Q^r:\mathcal{C}\to\mathcal{C}[S^{-1}]^r\).

Proof. By duality, we need only consider a left multiplicative system.

Theorem (P. Gabriel, M. Zisman). Let \(S\) be a left (or right) multiplicative system in \(\mathcal{C}\). Then \(\left(\mathcal{C}[S^{-1}]^l,Q^l\right)\) (or \(\left(\mathcal{C}[S^{-1}]^r,Q^r\right)\)) is a localization of \(\mathcal{C}\) at \(S\).

Now let \(F:\mathcal{C}\to\mathcal{D}\) be a functor sending \(S\) to isomorphisms. Define \(F[S^{-1}]:\mathcal{C}[S^{-1}]^l\to\mathcal{D}\) as follows: on objects it sends \(X\) to \(FX\), and on morphisms it sends \([U;s,a]\in\operatorname{Hom}_{\mathcal{C}[S^{-1}]^l}(X,Y)\) to \((Fa)(Fs)^{-1}\in\operatorname{Hom}_{\mathcal{D}}(FX,FY)\)³. Clearly, \(F[S^{-1}]\) sends identity morphisms in \(\mathcal{C}[S^{-1}]^l\) to identity morphisms in \(\mathcal{D}\). Apply \(F\) to the diagram in Definition–Proposition reference and invert the arrows in \(F(S)\); it follows at once that \(F[S^{-1}]\) preserves composition. The equality \(F[S^{-1}]Q=F\) is also clear. Since every morphism in \(\mathcal{C}[S^{-1}]^l\) has the form \((Qa)(Qs)^{-1}\), no other definition of \(F[S^{-1}]\) is possible.

Convention. If \(S\) is a multiplicative system, Proposition reference shows that the two localizations can be identified; we write the resulting functor as \(Q:\mathcal{C}\to\mathcal{C}[S^{-1}]\).

Corollary. Let \(S\) be a left (or right) multiplicative system in \(\mathcal{C}\). Morphisms \(f,g:X\to Y\) in \(\mathcal{C}\) satisfy \(Q^lf=Q^lg\) (or \(Q^rf=Q^rg\)) if and only if there exists \(s\in S\) such that \(fs=gs\) (or \(sf=sg\)).

Corollary. Let \(S\) be a left (or right) multiplicative system in \(\mathcal{C}\). Then \(Q^l\) (or \(Q^r\)) sends monomorphisms to monomorphisms (or epimorphisms to epimorphisms).

Proof. We prove only the case of a left multiplicative system. Let \(f:X\to Y\) be a monomorphism in \(\mathcal{C}\). Take \(\alpha,\beta:Q^lW\rightrightarrows Q^lX\) such that \((Q^lf)\alpha=(Q^lf)\beta\). Conditions (S2) and (S3) allow us to represent \(\alpha\) and \(\beta\) by elements \((U;s,a)\) and \((U;s,b)\), respectively, of \(M^l_{W,X}\); the details are left to the reader as an exercise. The given equality then implies \(Q^l(fa)=Q^l(fb)\). By Corollary reference, there is a \(t\in S\) such that \(fat=fbt\). Hence \(at=bt\), and therefore \(\alpha=\beta\).

Remark.

In the basic operations of category theory, we want to avoid large categories whenever possible. Definition–Theorem reference says that \(\mathcal{C}[S^{-1}]^l\) (or \(\mathcal{C}[S^{-1}]^r\)) may be a large category. The reason is that the \(\varinjlim\) in Lemma reference need not be small, so its value need not lie in \(\ensuremath{\mathsf{Set}}\). This is a somewhat delicate issue.

If \(S\) satisfies the following condition, however, then \(\mathcal{C}[S^{-1}]^l\) (or \(\mathcal{C}[S^{-1}]^r\)) is not large: assume that \(S_{/X}^{\operatorname{op}}\) (or \(S_{X/}\)) has a small cofinal subcategory (Definition reference) for every \(X\in\operatorname{Ob}(\mathcal{C})\). By Proposition reference, the colimit in question may be restricted to that subcategory, yielding an object of \(\ensuremath{\mathsf{Set}}\). If \(\mathcal{C}\) itself is small, this condition holds automatically.

Lemma. Let \(S\) be a left (or right) multiplicative system in \(\mathcal{C}\). Then the localization functor \(Q^l\) (or \(Q^r\)) preserves finite \(\varprojlim\) (or finite \(\varinjlim\)).

Proof. We discuss only the case of a left multiplicative system. Let \(J\) be a finite category, and suppose that the limit associated with a functor \(\beta:J^{\operatorname{op}}\to\mathcal{C}\) exists. By Lemma reference, for every \(X\in\operatorname{Ob}(\mathcal{C})\) there are canonical bijections \[\begin{multline*} \operatorname{Hom}_{\mathcal{C}[S^{-1}]^l}\left(Q^lX,Q^l\varprojlim\beta\right) \simeq \varinjlim_{X\leftarrowtail Z}\operatorname{Hom}_{\mathcal{C}}\left(Z,\varprojlim\beta\right) \\ \simeq \varinjlim_{X\leftarrowtail Z}\varprojlim_{j\in\operatorname{Ob}(J)} \operatorname{Hom}_{\mathcal{C}}\left(Z,\beta(j)\right). \end{multline*}\]

Applying Lemma reference and Proposition reference to the last term transforms it further into \[\varprojlim_{j \in \operatorname{Ob}(J)} \varinjlim_{X \leftarrowtail Z} \operatorname{Hom}_{\mathcal{C}}\left( Z, \beta(j) \right) \simeq \varprojlim_{j \in \operatorname{Ob}(J)} \operatorname{Hom}_{\mathcal{C}[S^{-1}]^l} \left(Q^l X, Q^l \beta(j) \right).\] Strictly speaking, §reference works in \(\ensuremath{\mathsf{Set}}\), whereas the \(\operatorname{Hom}\) sets here need not lie in \(\ensuremath{\mathsf{Set}}\). This is only a minor technical issue; see the related discussion in the proof of Lemma reference.

Theorem. Let \(\mathcal{C}\) be an \(\ensuremath{\mathsf{Ab}}\)-category and let \(S\) be a left (or right) multiplicative system in it.

1. The category \(\mathcal{C}[S^{-1}]^l\) (or \(\mathcal{C}[S^{-1}]^r\)) has a canonical \(\ensuremath{\mathsf{Ab}}\)-category structure making \(Q^l\) (or \(Q^r\)) an additive, essentially surjective functor.

2. If \(\mathcal{C}\) is an additive category and \(S\) is a multiplicative system, then \(\mathcal{C}[S^{-1}]\) is also an additive category.

3. The preceding statements remain true if \(\ensuremath{\mathsf{Ab}}\)-categories are replaced by \(\Bbbk\)-linear categories, where \(\Bbbk\) is a commutative ring. If \(F\) in the universal property of Definition reference is \(\Bbbk\)-linear, then so is the associated functor \(F[S^{-1}]\).

Proof. For (i), it suffices to treat a left multiplicative system. We must define addition on the morphism sets of \(\mathcal{C}[S^{-1}]\). Let \[f,g\in\operatorname{Hom}_{\mathcal{C}[S^{-1}]^l}(X,Y).\] Since \(S_{/X}^{\operatorname{op}}\) is filtered (Lemma reference), there are a morphism \(s:U\rightarrowtail X\) and morphisms \(a_1,a_2:U\to Y\) in \(\mathcal{C}\) such that \(f=[U;s,a_1]\) and \(g=[U;s,a_2]\). Define \[f+g:=[U;s,a_1+a_2]\in\operatorname{Hom}_{\mathcal{C}[S^{-1}]}(X,Y).\] In particular, a zero morphism can be written \([U;s,0]\), and \(-[U;s,a]=[U;s,-a]\). One must still prove that \(f+g\) is independent of the choice of representatives; this too follows from the filteredness of \(S_{/X}^{\operatorname{op}}\), and the routine details are omitted. Once this operation is known to be well defined, the axioms for an \(\ensuremath{\mathsf{Ab}}\)-category and the additivity of \(Q\) are easy to verify.

Statement (ii) follows from (i), the definition of an additive category (Definition reference), and Lemma reference.

For (iii), if \(\mathcal{C}\) is \(\Bbbk\)-linear, define scalar multiplication on \(\operatorname{Hom}_{\mathcal{C}[S^{-1}]}\) by \(t\cdot[U;s,a]=[U;s,ta]\) for \(t\in\Bbbk\). The required properties are immediate.

Example (Localization of rings). Giving a ring \(R\) is equivalent to giving an \(\ensuremath{\mathsf{Ab}}\)-category \(\mathcal{C}\) with a single object \(\star\) such that \(R=\operatorname{End}_{\mathcal{C}}(\star)\). If \(R\) is commutative, the conditions that \(S\subset R=\operatorname{Mor}(\mathcal{C})\) be a left or a right multiplicative system coincide. In this case, the ring corresponding to the \(\ensuremath{\mathsf{Ab}}\)-category \(\mathcal{C}[S^{-1}]\) is precisely the localization \(R[S^{-1}]\) of \(R\); see [Li1]. For a general ring \(R\), the theory in this section provides a construction of noncommutative localizations. The related theory is also a branch of noncommutative ring theory; see [Lam99] for a detailed account.

Example (Central localization). Write the center of an additive category \(\mathcal{C}\) as \(Z(\mathcal{C})\); this is a commutative ring. See the discussion preceding Proposition reference. For any multiplicative subset \(S\) of \(Z(\mathcal{C})\), it is easy to check that the set of morphisms \[\{s_X \in \operatorname{End}_{\mathcal{C}}(X): s \in S, \; X \in \operatorname{Ob}(\mathcal{C}) \}\] is a multiplicative system. The corresponding category \(\mathcal{C}[S^{-1}]\) can of course be constructed using the calculus of fractions above, but a shorter construction is \[\operatorname{Ob}(\mathcal{C}[S^{-1}]) := \operatorname{Ob}(\mathcal{C}), \quad \operatorname{Hom}_{\mathcal{C}[S^{-1}]}(X, Y) := \operatorname{Hom}_{\mathcal{C}}(X, Y) \ensuremath{\underset{Z(\mathcal{C})}{\otimes}} Z(\mathcal{C})[S^{-1}],\] with composition defined in the usual way. The relevant checks are left as an exercise for this chapter.

Remark.

For general \(\mathcal{C}\) and \(S\), the case in which the localization functor \(Q:\mathcal{C}\to\mathcal{C}[S^{-1}]\) has a fully faithful right adjoint is especially interesting and important. A localization with this property is called a reflective localization. The Gabriel–Popescu theorem to be introduced in Appendix § A.3 is a typical example.

Remark (Localization of product categories). Finally, let \(\mathcal{C}_i\) be categories equipped with left (or right) multiplicative systems \(S_i\), for \(i=1,2\), and write their localization functors as \(Q_i:\mathcal{C}_i\to\mathcal{C}_i[S_i^{-1}]\). Clearly, \(S_1\times S_2\) is also a left (or right) multiplicative system in \(\mathcal{C}_1\times\mathcal{C}_2\). From the construction in Definition–Theorem reference, one readily checks that \[(Q_1, Q_2): \mathcal{C}_1 \times \mathcal{C}_2 \to \mathcal{C}_1[S_1^{-1}] \times \mathcal{C}_2[S_2^{-1}]\] is the corresponding localization. This statement generalizes to products of arbitrary families of categories.

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1. From the perspective of \(2\)-categories, the more natural universal property is this: for every \(\mathcal{D}\), the functor \(Q^*:\mathcal{D}^{\mathcal{C}[S^{-1}]}\to\mathcal{D}^{\mathcal{C}}\) is fully faithful, and \(G\in\operatorname{Ob}(\mathcal{D}^{\mathcal{C}})\) is isomorphic to some \(Q^*(F)=FQ\) if and only if \(G\) sends the elements of \(S\) to isomorphisms. The functor \(Q\) in Definition reference automatically has this property—the fully faithful part is Proposition reference, and the rest is easy.↩︎

2. Translator’s note: the source has \(\ensuremath{\mathrm{id}}_{\mathcal{C}[S^{-1}]}\) for this last identity. Since \(GG'\) is an endofunctor of \(\mathcal{C}[S^{-1}]'\), the prime has been added to the target localization here (O014-C017).↩︎

3. Translator’s note: the source has \(\operatorname{Hom}_{\mathcal{D}}(X,Y)\). Since \((Fa)(Fs)^{-1}:FX\to FY\), the source and target objects have been corrected to \(FX\) and \(FY\) here (O014-C018).↩︎

Kan Extensions Along Localization

Proposition. Let \(\mathcal{I}\) be a full subcategory of \(\mathcal{C}\), let \(S\subset\operatorname{Mor}(\mathcal{C})\) be a left (or right) multiplicative system, and set \(T:=S\cap\operatorname{Mor}(\mathcal{I})\).

1. If \(T\) is assumed to be a left (or right) multiplicative system in \(\mathcal{I}\), these data induce a functor \(\mathcal{I}[T^{-1}]^l\to\mathcal{C}[S^{-1}]^l\) (or \(\mathcal{I}[T^{-1}]^r\to\mathcal{C}[S^{-1}]^r\)¹).

2. Assume the following condition: for every morphism \(s:W\to Y\) (or \(s:Y\to W\)) in \(S\) with \(Y\in\operatorname{Ob}(\mathcal{I})\), there is a morphism \(g:V\to W\) such that \(sg\in T\) (or \(g:W\to V\) such that \(gs\in T\)). Then \(T\) is a left (or right) multiplicative system, and the functor in (i) is fully faithful.

Proof. By duality, it suffices to consider a left multiplicative system. The functor in (i) comes from the universal property of localization; on morphisms it sends every equivalence class \([U;s,a]\) relative to \(T\) to the same class \([U;s,a]\) relative to \(S\).

Let \(X,Y\in\operatorname{Ob}(\mathcal{I})\). For a morphism in \(\mathcal{C}[S^{-1}]^l\) represented by \(X\leftarrowtail U\to Y\), the condition allows us to choose \(V\to U\) so that the composite \(V\to U\to X\) belongs to \(T\). Thus \(\operatorname{Hom}_{\mathcal{I}[T^{-1}]^l}(X,Y)\to \operatorname{Hom}_{\mathcal{C}[S^{-1}]^l}(X,Y)\) is surjective. A similar argument shows that every equivalence \(\sim\) in \(M_{X,Y}^l\) can be realized inside \(\mathcal{I}\), proving that the same map is injective.

The following lemma is stated only for a right multiplicative system \(S\); the case of a left multiplicative system is dual.

Lemma. Let \(S\subset\operatorname{Mor}(\mathcal{C})\) be a right multiplicative system. For a given \(\underline{X}:=QX\in \operatorname{Ob}\left(\mathcal{C}[S^{-1}]\right)\), define the category \((Q/\underline{X})\) as in Definition reference. We shall represent an object \((Y,QY\to\underline{X})\) of \((Q/\underline{X})\), nonuniquely, by a diagram \(Y\xrightarrow{a}U\stackrel{s}{\leftarrowtail}X\) in \(\mathcal{C}\).

Define \(S_{X/}\) as in §reference. The functor \(S_{X/}\to(Q/\underline{X})\) that sends an object \([X\stackrel{t}{\rightarrowtail}V]\) to the diagram \(V=V\stackrel{t}{\leftarrowtail}X\), which determines \((V,QV\to\underline{X})\), is cofinal (Definition reference).

Proof. For \([X\stackrel{t}{\rightarrowtail}V]\) as in the statement, specifying a morphism from \((Y,QY\to\underline{X})\) to the corresponding \((V,QV\to\underline{X})\) amounts to specifying a morphism \(b:Y\to V\) in \(\mathcal{C}\) such that the diagram \(Y\xrightarrow{b}V\stackrel{t}{\leftarrowtail}X\) represents the morphism \(QY\to\underline{X}\) in \(\mathcal{C}[S^{-1}]\).

Proposition. Let \(\mathcal{I}\) be a full subcategory of \(\mathcal{C}\), let \(S\subset\operatorname{Mor}(\mathcal{C})\) be a left (or right) multiplicative system, and put \(T:=S\cap\operatorname{Mor}(\mathcal{I})\). Assume the following “resolution condition”: for every \(X\in\operatorname{Ob}(\mathcal{C})\), there is a morphism \(s:U\to X\) (or \(s:X\to U\)), where \(U\in\operatorname{Ob}(\mathcal{I})\) and \(s\in S\).

1. Then \(T\subset\operatorname{Mor}(\mathcal{I})\) is a left (or right) multiplicative system, and \(\mathcal{I}[T^{-1}]\to\mathcal{C}[S^{-1}]\) is an equivalence.

2. Suppose further that a functor \(F:\mathcal{C}\to\mathcal{E}\) sends the elements of \(T\) to isomorphisms. Then \(\operatorname{Ran}_QF\) (or \(\operatorname{Lan}_QF\)) exists, and the following diagram commutes up to isomorphism: \[\text{Diagram chapter1-unit-018-d005}\] Here \(\mathcal{I}[T^{-1}]\to\mathcal{E}\) is determined by the universal property of localization.

3. For \(X\in\operatorname{Ob}(\mathcal{C})\), define the categories \(S_{/X}\) and \(S_{X/}\) as in §reference. Under the assumptions of (ii), there are canonical isomorphisms² \[\begin{align*} \left(\operatorname{Ran}_Q F\right)(QX) &\simeq \varprojlim_{[X \leftarrowtail Y] \in \operatorname{Ob}(S_{/X}^{\operatorname{op}})} FY, \\ \left(\operatorname{Lan}_Q F\right)(QX) &\simeq \varinjlim_{[X \rightarrowtail Y] \in \operatorname{Ob}(S_{X/})} FY. \end{align*}\]

The canonical morphism of the Kan extension \(\varepsilon:(\operatorname{Ran}_QF)Q\to F\) (or \(\eta:F\to(\operatorname{Lan}_QF)Q\)) is, in the situation of (iii), determined by the term \(FX\) corresponding to \(\ensuremath{\mathrm{id}}_X\in\operatorname{Ob}(S_{/X}^{\operatorname{op}})\) (or \(\ensuremath{\mathrm{id}}_X\in\operatorname{Ob}(S_{X/})\)).

Proof. It suffices to discuss a right multiplicative system. The condition in Proposition reference(ii) plainly holds, so \(T\) is a right multiplicative system. Denote its localization by \(Q':\mathcal{I}\to\mathcal{I}[T^{-1}]\), and denote the fully faithful functor \(\mathcal{I}[T^{-1}]\to\mathcal{C}[S^{-1}]\) of Proposition reference(ii) by \(\iota_Q\). The resolution condition also ensures that \(\iota_Q\) is essentially surjective, hence an equivalence. This proves (i).

Choose any quasi-inverse \(\iota_Q^{-1}\) to \(\iota_Q\). We will show that \(F'\circ\iota_Q^{-1}\) gives the left Kan extension \(\operatorname{Lan}_QF\); equivalently, \(\operatorname{Lan}_QF\) exists and \(F'\simeq(\operatorname{Lan}_QF)\circ\iota_Q\).

To prove the existence of \(\operatorname{Lan}_QF\), define \(\Pi_{/\underline{X}}:(Q/\underline{X})\to\mathcal{C}\) as in Definition reference, where \(\underline{X}:=QX\in\operatorname{Ob}(\mathcal{C}[S^{-1}])\). In view of Theorem reference(i), it is enough to show that \(\varinjlim F\Pi_{/\underline{X}}\) exists for every \(\underline{X}\).

Apply the cofinal functor \(S_{X/}\to(Q/\underline{X})\) of Lemma reference and Proposition reference. This reduces \(\varinjlim F\Pi_{/\underline{X}}\) to \(\varinjlim_{[X\rightarrowtail U]}FU\), with the colimit taken over \(S_{X/}\).

Since \(S_{X/}\) is filtered (Lemma reference), the resolution condition together with Proposition reference ensures that the objects \([X\rightarrowtail U]\) with \(U\in\operatorname{Ob}(\mathcal{I})\) form a cofinal full subcategory of \(S_{X/}\). Another application of Proposition reference therefore lets us restrict the colimit to \(U\in\operatorname{Ob}(\mathcal{I})\).

Finally consider (iii). For \(X\in\operatorname{Ob}(\mathcal{C})\), the preceding argument shows that \(\varinjlim_{[X\rightarrowtail Y]\in\operatorname{Ob}(S_{X/})}FY\) exists and gives \((\operatorname{Lan}_QF)(QX)\). Under this correspondence, the canonical morphism \(\eta_X\) of the left Kan extension is naturally determined by the term corresponding to \([X\stackrel{\ensuremath{\mathrm{id}}}{\rightarrowtail}X]\); see the construction in §reference.

Proposition. Under the hypotheses of Proposition reference, \(\operatorname{Ran}_QF\) (or \(\operatorname{Lan}_QF\)) is the absolute right (or left) Kan extension of \(F\) along \(Q\); see Definition reference.

Proof. The first condition in Proposition reference concerns only the left (or right) multiplicative system \(S\) in \(\mathcal{C}\) and the subcategory \(\mathcal{I}\), not \(F\); the second only requires \(F\) to send \(T:=S\cap\operatorname{Mor}(\mathcal{I})\) to isomorphisms. In particular, for any functor \(M:\mathcal{E}\to\mathcal{F}\), the condition on \(F\) is inherited at once by \(MF:\mathcal{C}\to\mathcal{F}\).

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1. Translator’s note: in the right-hand case, the source gives the target as \(\mathcal{C}[T^{-1}]^r\). It has been corrected to \(\mathcal{C}[S^{-1}]^r\), since \(T\subset\operatorname{Mor}(\mathcal{I})\) whereas \(S\subset\operatorname{Mor}(\mathcal{C})\) (O014-C019).↩︎

2. The limit in the first formula is indexed by \(\operatorname{Ob}(S_{/X}^{\operatorname{op}})\) because we use \(\varprojlim_i\beta(i)\) to denote the limit of a functor \(\beta:I^{\operatorname{op}}\to\mathcal{C}\).↩︎

Adjoint Functor Theorems

Our first task is to reformulate the existence of an adjoint appropriately. By duality, we state only the case of a left adjoint. For a functor \(G:\mathcal{D}\to\mathcal{C}\) and \(c\in\operatorname{Ob}(\mathcal{C})\), Definition reference gives a category \((c/G)\) whose objects are data \((d,c\xrightarrow{f}Gd)\).

Lemma. A functor \(G:\mathcal{D}\to\mathcal{C}\) has a left adjoint if and only if \((c/G)\) has an initial object for every \(c\in\operatorname{Ob}(\mathcal{C})\).

Suppose that \(G\) has a left adjoint \(F\), and let \(c\in\operatorname{Ob}(\mathcal{C})\). We claim that \(Fc\), together with the unit \(\eta_c:c\to G(Fc)\), is initial in \((c/G)\). For any \((d,c\xrightarrow{\beta}Gd)\), the map in equation, \(\operatorname{Hom}_{\mathcal{D}}(Fc,d)\to\operatorname{Hom}_{\mathcal{C}}(c,Gd)\), sends \(\alpha\) to \((G\alpha)\eta_c\). This is exactly the bijection on \(\operatorname{Hom}\) sets supplied by the adjunction, so \((Fc,\eta_c)\) is initial.

Conversely, suppose that \((c/G)\) has an initial object for every \(c\), and write it as \((Fc,\eta_c:c\to G(Fc))\). Since initial objects are unique up to unique isomorphism, these objects form a functor \(F:\mathcal{C}\to\mathcal{D}\), and \((\eta_c)_{c\in\operatorname{Ob}(\mathcal{C})}\) gives \(\eta:\ensuremath{\mathrm{id}}_{\mathcal{C}}\to GF\). For every \((c,d)\in\operatorname{Ob}(\mathcal{C})\times\operatorname{Ob}(\mathcal{D})\), define the bijection from equation: \[\begin{align*} \varphi_{c, d}: \operatorname{Hom}_{\mathcal{D}}(Fc, d) & \to \operatorname{Hom}_{\mathcal{C}}(c, Gd) \\ \alpha & \mapsto (G\alpha) \eta_c. \end{align*}\] These bijections are plainly functorial in \(c\) and \(d\). Hence \((F,G,\varphi)\) is an adjoint pair with unit \(\eta\).

Definition. Let \(\mathcal{E}\) be a category. A nonempty small subset \(\Gamma\) of \(\operatorname{Ob}(\mathcal{E})\) is called a weakly initial family if, for every \(e'\in\operatorname{Ob}(\mathcal{E})\), there is an \(e\in\Gamma\) such that \(\operatorname{Hom}_{\mathcal{E}}(e,e')\) is nonempty.¹ If \(e\in\operatorname{Ob}(\mathcal{E})\) and \(\{e\}\) is weakly initial, then \(e\) itself is called a weakly initial object.

Every initial object is weakly initial. Weakly terminal families and weakly terminal objects are defined similarly.

Lemma. If \(\Gamma\) is a weakly initial family in \(\mathcal{E}\) and \(e:=\prod_{x\in\Gamma}x\) exists, then \(e\) is weakly initial.

Proof. For every \(e'\in\operatorname{Ob}(\mathcal{E})\), there are \(y\in\Gamma\) and a morphism \(y\to e'\). Take the composite \(e\xrightarrow{\text{projection}}y\to e'\).

Return to adjoint functor theorems. To apply Lemma reference, we must study the projection functor \(\Pi_{c/}:(c/G)\to\mathcal{D}\) from Definition reference.

Proposition. Let \(\mathcal{D}\) be complete, let \(G:\mathcal{D}\to\mathcal{C}\) preserve all small limits, and fix \(c\in\operatorname{Ob}(\mathcal{C})\).

1. The functor \(\Pi_{c/}\) is conservative (Definition reference) and creates all small limits (Definition reference).

2. The category \((c/G)\) is complete.

3. The functor \(\Pi_{c/}\) sends monomorphisms to monomorphisms.

Statement (ii) follows from the completeness of \(\mathcal{D}\) and (i).

Now consider a monomorphism in \((c/G)\) from \((d',c\to Gd')\) to \((d,c\to Gd)\), determined by \(\iota:d'\to d\) in \(\mathcal{D}\). By (i), \[(d', c \to Gd') \ensuremath{\underset{(d, c \to Gd)}{\times}} (d', c \to Gd') = \left( d' \ensuremath{\underset{d}{\times}} d' , \; c \to G\left( d' \ensuremath{\underset{d}{\times}} d' \right) \right) .\] The left-hand side identifies with \((d',c\to Gd')\), with both projections corresponding to \(\ensuremath{\mathrm{id}}_{d'}\). Thus \(d'\ensuremath{\underset{d}{\times}}d'\) identifies with \(d'\) in the same way. In other words, \(\iota\) is monic.

Lemma. If a complete category \(\mathcal{E}\) has a weakly initial family \(\Gamma\), then \(\mathcal{E}\) has an initial object.

We show that \(x\) is initial. Given \(y\in\operatorname{Ob}(\mathcal{E})\), choose any morphism \(e\to y\) and call the composite \(x\xrightarrow{i}e\to y\) by \(f\). For any \(g:x\to y\), consider the equalizer \(j:\operatorname{ker}(f,g)\hookrightarrow x\). There is a morphism \(k:e\to\operatorname{ker}(f,g)\), hence \(ijk:e\to e\). Taking \(\psi=\ensuremath{\mathrm{id}}_x\) in equation gives \((ijk)i=(\ensuremath{\mathrm{id}}_e)i=i\). Cancelling the monomorphism \(i\) on the left gives \(jki=\ensuremath{\mathrm{id}}_x\), and consequently \(f=fjki=gjki=g\). This proves the claim.

The following result is due to P. Freyd and is also known as the General Adjoint Functor Theorem.

Theorem (P. Freyd’s Adjoint Functor Theorem).

Let \(\mathcal{C}\) and \(\mathcal{D}\) be categories.

1. Let \(\mathcal{D}\) be complete and let \(G:\mathcal{D}\to\mathcal{C}\) preserve all small limits. Then \(G\) has a left adjoint if and only if the following solution set condition holds: for every \(c\in\operatorname{Ob}(\mathcal{C})\), there is a family of morphisms \((f_i:c\to Gd_i)_{i\in I}\), indexed by a small set \(I\), such that for every \(f:c\to Gd\) there are \(i\in I\) and \(\alpha:d_i\to d\) for which \(f\) factors as \(c\xrightarrow{f_i}Gd_i\xrightarrow{G\alpha}Gd\).

2. Let \(\mathcal{C}\) be cocomplete and let \(F:\mathcal{C}\to\mathcal{D}\) preserve all small colimits. Then \(F\) has a right adjoint if and only if the following cosolution set condition holds: for every \(d\in\operatorname{Ob}(\mathcal{D})\), there is a family of morphisms \((g_i:Fc_i\to d)_{i\in I}\), indexed by a small set \(I\), such that for every \(g:Fc\to d\) there are \(i\in I\) and \(\beta:c\to c_i\) for which \(g\) factors as \(Fc\xrightarrow{F\beta}Fc_i\xrightarrow{g_i}d\).

Proof. We prove only (i). Fix \(c\in\operatorname{Ob}(\mathcal{C})\). By Lemma reference, \(G\) has a left adjoint if and only if \((c/G)\) has an initial object. The solution set condition says exactly that \((c/G)\) has the weakly initial family \(\{(d_i,c\to Gd_i):i\in I\}\), where \(I\) is small. Since every initial object is weakly initial, the “only if” direction is clear. In the other direction, Lemma reference reduces the problem to the completeness of \((c/G)\), which is precisely Proposition reference(ii).

Example. Consider the category of groups \(\ensuremath{\mathsf{Grp}}\) and the forgetful functor \(U:\ensuremath{\mathsf{Grp}}\to\ensuremath{\mathsf{Set}}\). The category \(\ensuremath{\mathsf{Grp}}\) is complete and \(U\) preserves all small limits. To deduce from Theorem reference that \(U\) has the left adjoint \(\mathrm{F}:\ensuremath{\mathsf{Set}}\to\ensuremath{\mathsf{Grp}}\), which constructs the free group on a set, the key is to verify the solution set condition for every small set \(c\).

Take any \(d\in\operatorname{Ob}(\ensuremath{\mathsf{Grp}})\) and map \(f:c\to Ud\). Factor \(f\) as \(c\to Us\xrightarrow{U\iota}Ud\), where \(s\) is the subgroup of \(d\) generated by \(f(c)\) and \(\iota:s\to d\) is the inclusion. Presenting \(s\) by the generating set \(f(c)\) and relations shows that the isomorphism classes of all such data \((s,c\to Us)\) form a small set \(I\). This establishes the solution set condition.

The same procedure works for other algebraic structures, such as \(\ensuremath{\mathsf{Ab}}\) and \(R\ensuremath{\text{-}\mathsf{Mod}}\), again giving left adjoints to the forgetful functors; see the latter part of [ML98]. Likewise, for any commutative ring \(\Bbbk\), taking the multiplicative monoid of a \(\Bbbk\)-algebra gives a forgetful functor \(U:\Bbbk\ensuremath{\text{-}\mathsf{Alg}}\to\ensuremath{\mathsf{Mon}}\). The same technique shows that \(U\) has a left adjoint sending a monoid \(M\) to \(\Bbbk[M]\); see [Li1].

The approach through the Adjoint Functor Theorem reference not only applies uniformly to many algebraic structures; in the free-group case it is also shorter than the classical construction in [Li1].

The Special Adjoint Functor Theorem reference, to be introduced shortly, replaces the solution set condition by conditions on generators and subobjects. We pause to define generators and cogenerators in a category; these notions will also be used in the later section on Grothendieck categories, among other places.

Definition.

Let \(\mathcal{E}\) be a category and let \(\Sigma\) be a nonempty small subset of \(\operatorname{Ob}(\mathcal{E})\).

• The set \(\Sigma\) is called a generating family if the following holds: given any pair \(f,g:x\to y\) in \(\mathcal{E}\), if \(f\epsilon=g\epsilon\) for every \(s\in\Sigma\) and every \(\epsilon:s\to x\), then \(f=g\). If \(\{s\}\) is a generating family for \(\mathcal{E}\), then \(s\) is called a generator of \(\mathcal{E}\).

• The set \(\Sigma\) is called a cogenerating family if the following holds: given any pair \(f,g:x\to y\) in \(\mathcal{E}\), if \(\delta f=\delta g\) for every \(s\in\Sigma\) and every \(\delta:y\to s\), then \(f=g\). If \(\{s\}\) is a cogenerating family for \(\mathcal{E}\), then \(s\) is called a cogenerator of \(\mathcal{E}\).

The two notions are plainly dual.

Lemma. Let \(\Sigma\) be a small subset of \(\operatorname{Ob}(\mathcal{C})\). If every object of \(\mathcal{C}\) can be expressed as a colimit (or limit) of elements of \(\Sigma\), then \(\Sigma\) is a generating (or cogenerating) family.

Proof. For colimits, a morphism \(\varinjlim\alpha=:x\to y\) is uniquely determined by its composites with all \(\alpha(i)\xrightarrow{\iota_i}\varinjlim\alpha\).

Example. Here are some basic examples of generators and cogenerators.

• In \(\ensuremath{\mathsf{Set}}\), the singleton \(\{\mathrm{pt}\}\) is a generator: for maps \(f,g:X\to Y\), specifying \(\epsilon:\{\mathrm{pt}\}\to X\) amounts to specifying \(x\in X\), so \(f\epsilon=g\epsilon\) for every \(\epsilon\) is equivalent to pointwise equality of \(f\) and \(g\). The set \(\{0,1\}\) is a cogenerator, because if \(f(x)\ne g(x)\) for some \(x\in X\), one can choose \(\delta:Y\to\{0,1\}\) such that \(\delta(f(x))\ne\delta(g(x))\).

• Let \(\ensuremath{\mathsf{CHaus}}\) be the category of compact Hausdorff spaces and continuous maps. The singleton \(\{\mathrm{pt}\}\) is again a generator, while \([0,1]\) is a cogenerator: for every compact Hausdorff space \(Y\) and distinct \(a,b\in Y\), Urysohn’s lemma [Xiong] gives a continuous map \(\delta:Y\to[0,1]\) with \(\delta(a)=0\) and \(\delta(b)=1\).

• Let \(R\) be a ring. In \(R\ensuremath{\text{-}\mathsf{Mod}}\), the module \(R\) is a generator because specifying an \(R\)-module homomorphism \(R\to M\) amounts to specifying an element of \(M\). Taking coproducts shows that every nonzero free \(R\)-module is also a generator.

• Consider the Yoneda embedding \(h_{\mathcal{E}}:\mathcal{E}\to\mathcal{E}^\wedge\) of a small category \(\mathcal{E}\). Then \(h_{\mathcal{E}}(\operatorname{Ob}(\mathcal{E}))\) is a generating family for \(\mathcal{E}^\wedge\). This follows immediately from the density theorem for the Yoneda embedding in Appendix § A.1 and Lemma reference.

The Special Adjoint Functor Theorem involves one more set-theoretic notion. Recall the posets \((\mathrm{Sub}_e,\subset)\) and \((\mathrm{Quot}_e,\twoheadleftarrow)\) introduced in Definition reference, where \(e\) is an object of a category \(\mathcal{E}\).

Definition.

A category \(\mathcal{E}\) is called well-powered (or well-copowered) if \(\mathrm{Sub}_e\) (or \(\mathrm{Quot}_e\)) is a small set for every \(e\in\operatorname{Ob}(\mathcal{E})\).

In this section, well-poweredness will be paired with completeness. Choose a representative \(s\hookrightarrow e\) of every element of \(\mathrm{Sub}_e\). Thus for any subset \(S\subset\mathrm{Sub}_e\) it makes sense, independently of the representatives chosen, to ask whether the fiber product \(\prod_{s\in S}(s\hookrightarrow e)\) exists. Dually, for a subset \(S\subset\mathrm{Quot}_e\), one may ask whether the fiber coproduct \(\coprod_{s\in S}(e\twoheadrightarrow s)\) exists.

• If \(\mathcal{E}\) is complete, then well-poweredness implies that for every \(e\), all subsets \(S\subset\mathrm{Sub}_e\) have fiber products.

• If \(\mathcal{E}\) is cocomplete, then well-copoweredness implies that for every \(e\), all subsets \(S\subset\mathrm{Quot}_e\) have fiber coproducts.

Lemma. Let \(\mathcal{E}\) be complete and well-powered. If there is a cogenerating family \(\Sigma\subset\operatorname{Ob}(\mathcal{E})\), then \(\mathcal{E}\) has an initial object.

Proof. Take \(e:=\prod_{s\in\Sigma}s\), and let \(x\) be the fiber product of all elements of \(\mathrm{Sub}_e\); only here is well-poweredness used. We show that \(x\) is initial in \(\mathcal{E}\).

By construction, \(x\hookrightarrow e\) is the infimum of the poset \((\mathrm{Sub}_e,\subset)\); in particular, \(\mathrm{Sub}_x=\{x\}\). For any object \(y\) and \(f,g\in\operatorname{Hom}_{\mathcal{E}}(x,y)\), the equalizer \(\operatorname{ker}(f,g)\) must therefore be \(x\) itself, so \(f=g\). It remains to show that \(\operatorname{Hom}_{\mathcal{E}}(x,y)\) is nonempty for every \(y\).

Theorem (Special Adjoint Functor Theorem).

Let \(\mathcal{C}\) and \(\mathcal{D}\) be categories.

1. A functor \(G:\mathcal{D}\to\mathcal{C}\) has a left adjoint if it satisfies the following conditions.

   • \(\mathcal{D}\) is complete and well-powered, and \(G\) preserves all small limits;

   • there is a cogenerating family \(\Sigma\subset\operatorname{Ob}(\mathcal{D})\).

2. A functor \(F:\mathcal{C}\to\mathcal{D}\) has a right adjoint if it satisfies the following conditions.

   • \(\mathcal{C}\) is cocomplete and well-copowered, and \(F\) preserves all small colimits;

   • there is a generating family \(\Sigma\subset\operatorname{Ob}(\mathcal{C})\).

Proof. By duality, it suffices to prove (i). Lemma reference again reduces the problem to showing that \((c/G)\) has an initial object for \(c\in\operatorname{Ob}(\mathcal{C})\); the strategy is to apply Lemma reference.

Proposition reference(ii) shows that \((c/G)\) is complete. Moreover, Definition reference and the definition of \((c/G)\) show at once that all data \((d,c\to Gd)\) with \(d\in\Sigma\) form a cogenerating family for \((c/G)\). Since \(\Sigma\) is small, this is a small set.

Let \(e=(d,c\to Gd)\in\operatorname{Ob}((c/G))\). Subobjects are isomorphism classes of monomorphisms. By Proposition reference, the projection \(\Pi_{c/}:(c/G)\to\mathcal{D}\) is conservative and preserves monomorphisms, so it induces an order-preserving embedding \(\mathrm{Sub}_e\hookrightarrow\mathrm{Sub}_d\). Thus \((c/G)\) is well-powered.

Corollary. If a complete well-powered category \(\mathcal{D}\) has a cogenerating family \(\Sigma\), then \(\mathcal{D}\) is cocomplete.

Dually, if a cocomplete well-copowered category \(\mathcal{C}\) has a generating family \(\Sigma\), then \(\mathcal{C}\) is complete.

Proof. It suffices to prove the assertion for \(\mathcal{D}\). Let \(I\) be a small category. By Example reference, it is enough to show that \((\alpha/\Delta)\) has an initial object for every \(\alpha:I\to\mathcal{D}\). By Lemma reference, this is equivalent to saying that the diagonal functor \(\Delta\) has a left adjoint. Clearly \(\Delta\) preserves all small limits and colimits, so Theorem reference applies.

Example. Let \(\iota:\ensuremath{\mathsf{CHaus}}\to\ensuremath{\mathsf{Top}}\) be the inclusion of compact Hausdorff spaces into topological spaces. Example reference showed that \(\ensuremath{\mathsf{CHaus}}\) has the cogenerator \([0,1]\). By the Tychonoff theorem [Xiong], \(\ensuremath{\mathsf{CHaus}}\) is readily seen to be complete, and \(\iota\) preserves all small limits; details are left to the reader. Furthermore, the power set of a small set is small, so \(\ensuremath{\mathsf{CHaus}}\) is well-powered. Theorem reference(i) therefore gives a left adjoint \(C:\ensuremath{\mathsf{Top}}\to\ensuremath{\mathsf{CHaus}}\) to \(\iota\). In point-set topology, \(C(X)\) is called the Stone–Čech compactification of \(X\); the unit gives a canonical morphism \(\eta_X:X\to\iota C(X)\). The constructions in Lemma reference and Theorem reference agree with the classical construction of the Stone–Čech compactification.

Adjoint functor theorems provide sufficient conditions for representability. For representable functors, see [Li1] or the definition of a representable functor in Appendix § A.1. The key observation is this: if \(G:\mathcal{D}\to\ensuremath{\mathsf{Set}}\) has a left adjoint \(F\), then \(G\) is representable. Indeed, consider the singleton \(\{\mathrm{pt}\}\) and put \(r_G:=F(\{\mathrm{pt}\})\). The canonical bijections \[\operatorname{Hom}_{\mathcal{D}}\left( r_G, d \right) \simeq \operatorname{Hom}_{\ensuremath{\mathsf{Set}}}\left( \{\mathrm{pt}\}, Gd \right) \simeq Gd , \quad d \in \operatorname{Ob}(\mathcal{D})\] imply \(G\simeq\operatorname{Hom}_{\mathcal{D}}(r_G,\cdot)\).

Corollary. If a complete well-powered category \(\mathcal{D}\) has a cogenerating family \(\Sigma\), then a functor \(G:\mathcal{D}\to\ensuremath{\mathsf{Set}}\) is representable if and only if it preserves all small limits.

Proof. The “only if” direction is a general property of representable functors. For the converse, Theorem reference ensures that \(G\) has a left adjoint \(F:\ensuremath{\mathsf{Set}}\to\mathcal{D}\); now apply the observation above.

Exercises

Consider the category of groups \(\ensuremath{\mathsf{Grp}}\). Prove that a group homomorphism \(\varphi:H\to G\) is a monomorphism (or epimorphism) in \(\ensuremath{\mathsf{Grp}}\) if and only if its underlying map is injective (or surjective). Some references, including [Li1], do not adequately explain this fact. Hint. It suffices to prove the “only if” direction. For monomorphisms, consider homomorphisms from \(\operatorname{ker}\varphi\) to \(H\). For epimorphisms, one must show that if \(\varphi(H)\ne G\), then there are distinct group homomorphisms \(f,g:G\rightrightarrows K\) whose restrictions to \(\varphi(H)\) agree. One approach is to take \(K\) to be the amalgamated free product defined by two copies of the embedding \(\varphi(H)\hookrightarrow G\) in [Li1], and to take \(f\) and \(g\) as the embeddings of \(G\) into the two copies. Uniqueness of reduced expressions [Li1] implies \(f\ne g\).

Show that in the category \(\ensuremath{\mathsf{Top}}\), the image of \(f:X\to Y\) is \(f(X)\) with the subspace topology from \(Y\), while its coimage is \(f(X)\) with the quotient topology from \(X\). Give the corresponding topological interpretations of strict monomorphisms and strict epimorphisms.

Let \(\mathcal{C}\) have finite limits and finite colimits. Prove that the following statements about a morphism \(f:X\to Y\) are equivalent.

1. The morphism \(f\) is a strict epimorphism.

2. The canonical morphism \(\operatorname{coim}(f)\to Y\) is an isomorphism.

3. \(X\ensuremath{\underset{Y}{\times}}X\rightrightarrows X\xrightarrow{f}Y\) is a coequalizer diagram.

4. The morphism \(f:X\to Y\) is the coequalizer of some pair \(a,b:W\rightrightarrows X\).

Let \(\mathcal{C}\) have finite limits and finite colimits. Prove that if a morphism \(f\) has a right inverse (or left inverse), then \(f\) is a strict epimorphism (or strict monomorphism). Hint. If \(fg=\ensuremath{\mathrm{id}}_Y\), then \(f\) is the coequalizer of \(gf\) and \(\ensuremath{\mathrm{id}}_X\).

Prove that an additive-category structure on any category, if it exists, is unique. Hint. Apply Proposition reference.

Compare Example reference. Let \(U\) be the forgetful functor from compact Hausdorff spaces \(\ensuremath{\mathsf{CHaus}}\) to sets \(\ensuremath{\mathsf{Set}}\). Prove that \(U\) creates all small limits. Show that it does not create all small colimits.

Let \(\mathcal{I}\) be the category of all finite sets and surjective maps. Prove that \(\mathcal{I}^{\operatorname{op}}\) is not filtered in the sense of §reference.

In Definition reference, prove that every morphism \(F\to F'\) induces canonical morphisms \(\operatorname{Lan}_KF\to\operatorname{Lan}_KF'\) and \(\operatorname{Ran}_KF\to\operatorname{Ran}_KF'\), provided these Kan extensions exist. Characterize the induced morphisms.

Let \(K\) and \(F\) be as in §reference. Prove that if \(M:\mathcal{E}\to\mathcal{F}\) has a left adjoint (or right adjoint), then \(M\) preserves \(\operatorname{Ran}_KF\) (or \(\operatorname{Lan}_KF\)). Hint. By duality, suppose that \(M\) has a right adjoint \(N\), with unit \(\eta':\ensuremath{\mathrm{id}}\to NM\) and counit \(\varepsilon':MN\to\ensuremath{\mathrm{id}}\). For any functor \(L:\mathcal{D}\to\mathcal{F}\), there are canonical bijections \[\begin{multline*} \operatorname{Hom}_{\mathcal{F}^{\mathcal{D}}} \left( M (\operatorname{Lan}_K F) , L \right) \simeq \operatorname{Hom}_{\mathcal{E}^{\mathcal{D}}} \left( \operatorname{Lan}_K F, N L \right) \\ \stackrel{\text{\href{#eqn:Kan-univ}{equation}}}{\simeq} \operatorname{Hom}_{\mathcal{E}^{\mathcal{C}}} \left( F, NLK \right) \simeq \operatorname{Hom}_{\mathcal{F}^{\mathcal{C}}} \left( MF, LK \right). \end{multline*}\]

Using the general properties of \(\eta'\) and \(\varepsilon'\), verify that when \(L=M(\operatorname{Lan}_KF)\) this bijection sends \(\ensuremath{\mathrm{id}}\) to \(M\eta\), where \(\eta\) is the morphism belonging to \(\operatorname{Lan}_KF\).

For general \(L\), use the familiar technique from the proof of the Yoneda lemma to verify that an arbitrary morphism \[\varphi:M(\operatorname{Lan}_KF)\to L\] corresponds to the composite \[\begin{aligned} (\varphi K)(M\eta):\quad MF &\xrightarrow{M\eta} M(\operatorname{Lan}_K F)K, \\ M(\operatorname{Lan}_K F)K &\xrightarrow{\varphi K} LK . \end{aligned}\]

1. The functor \(G\) is fully faithful.

2. The counit \(\varepsilon:FG\to\ensuremath{\mathrm{id}}_{\mathcal{D}}\) is an isomorphism.

3. The functor \(H\) is an equivalence of categories, with \(QG\) as a quasi-inverse.

4. For every category \(\mathcal{X}\), the functor \(F^*:\mathcal{X}^{\mathcal{D}}\to\mathcal{X}^{\mathcal{C}}\) is fully faithful.

Also state the dual version of this result. Hint. The following argument is from [GZ67]. First, (i) \(\iff\) (ii) is fairly easy; see the hint in [Li1].

For (ii) \(\implies\) (iii), one must prove \(QGH\simeq\ensuremath{\mathrm{id}}_{\mathcal{C}[S^{-1}]}\). Proposition reference reduces this to \(QGHQ\simeq Q\). Consider the unit \(\eta:\ensuremath{\mathrm{id}}_{\mathcal{C}}\to GF\). Since \((\varepsilon F)(F\eta)=\ensuremath{\mathrm{id}}_F\), the morphism \(F\eta\) is an isomorphism, hence \(Q\eta:Q\ensuremath{\stackrel{\sim}{\rightarrow}}QGF=QGHQ\).

For (iii) \(\implies\) (iv), the fact that \(H\) is an equivalence implies that \(H^*\) is fully faithful, while Proposition reference applies to \(Q^*\).

For (iv) \(\implies\) (ii), put \(E=\ensuremath{\mathrm{id}}_{\mathcal{D}}\) in the preceding exercise. Then \(FG\) is a right adjoint to \(\ensuremath{\mathrm{id}}_{\mathcal{D}}\), with counit \(\varepsilon\). Since right adjoints to \(\ensuremath{\mathrm{id}}_{\mathcal{D}}\) are unique up to isomorphism, \(\varepsilon\) is an isomorphism.

Verify the details of central localization in Example reference. Prove that the category \(\mathcal{C}[S^{-1}]\) defined there and the localization supplied by Theorem reference are equivalent as \(Z(\mathcal{C})[S^{-1}]\)-linear categories.

Complete the details in the proof of Proposition reference(i).

Use the Adjoint Functor Theorem reference to construct the free product \(G\star H\) of any two groups \(G\) and \(H\); see [Li1].

Let \(\mathrm{F}(X)\) be the free group on a set \(X\). Prove that the canonical map \(\iota:X\to U(\mathrm{F}(X))\) is injective, where \(U\) is the forgetful functor from groups to sets. Hint. For distinct \(x,y\in X\), there is a group \(H\) and a map of sets \(f:X\to U(H)\) such that \(f(x)\ne f(y)\).

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1. Translator’s note: the source has the subscript \(\mathcal{C}\); it has been corrected to \(\mathcal{E}\), the category introduced in this definition (O014-C020).↩︎

Abelian Categories

An abelian category is an additive category with kernels and cokernels in which every morphism is strict. Abelian categories generalize the category \(R\ensuremath{\text{-}\mathsf{Mod}}\) of left modules or \(\ensuremath{\mathsf{Mod}\text{-}}R\) of right modules over a ring \(R\), and support the study of such notions as injective and projective objects, complexes, homology and cohomology, and exact sequences. Module categories are the prototype of abelian categories, but do not exhaust their substance. There are two reasons for this.

1. In a general abelian category, objects have no elements and morphisms are not maps. Diagram-chasing methods from module categories are therefore no longer available; the most basic exact sequences must instead be derived by operations on objects and functors. Prominent examples are the snake lemma and the five lemma discussed in §reference, whose proofs use a substitute for diagram chasing (Lemma reference). Even so, the arguments are considerably more involved than their module-category counterparts.

2. Even when one starts with a module category or one of its abelian subcategories, abstract abelian categories arise when studying functor categories, localizations, or Serre quotients. In geometry and topology, sheaf theory naturally leads to such constructions.

The Freyd–Mitchell embedding theorem reference states that every small abelian category admits a fully faithful embedding into some \(\ensuremath{\mathsf{Mod}\text{-}}R\), and that the embedding functor preserves exact sequences. Once this result is accepted, many assertions about exact sequences reduce readily to the case of \(\ensuremath{\mathsf{Mod}\text{-}}R\). This reduction keeps diagram chasing available, but its chief benefit may be psychological reassurance: on the one hand it sidesteps the essence of abelian categories, and on the other hand any proof of the embedding theorem must still establish several core facts about abelian categories from scratch. This chapter does not depend on the embedding theorem, nor logically on module theory, but readers are expected to have a basic knowledge of homological algebra in module categories. Diagram-chasing skills remain useful and may even be indispensable in the natural order of study; readers are encouraged to consult [Li1] or the relevant parts of another textbook. Appendix §reference derives the embedding theorem as an application of the Ind construction. A cocomplete abelian category with a generator in which filtered small colimits are exact is called a Grothendieck category. Such categories are the subject of the final section, §reference; these conditions are directly related to assumptions on set size. Grothendieck categories occur frequently in geometry and enjoy better properties, such as completeness (Corollary reference) and the existence of canonical injective resolutions (Theorem reference). They are a closer analogue of module categories than general abelian categories; the precise meaning of this statement is explained by the Gabriel–Popescu theorem reference in the appendix. The basic isomorphism theorems of module theory remain valid for general abelian categories. The same is true of such fundamental notions as subobjects, quotient objects, direct-sum decompositions, indecomposable objects, simple objects, and composition series, all central concerns of algebra. They will be treated in §§reference— reference. For example, the Krull–Remak–Schmidt theorem on direct-sum decompositions has an abstract generalization (Theorem reference). The language of lattice theory is useful here; see §reference. It should be stressed that statements involving infinite direct sums often hold only in Grothendieck categories. Section reference begins with the definition and characterization of abelian subcategories, and then introduces quotients of an abelian category by a certain kind of abelian subcategory, called a Serre subcategory; the result is a Serre quotient. These quotients are characterized by a universal property and can be constructed by the localization introduced in §reference. The section also introduces the group \(\mathrm{K}_0\) of an abelian category and its basic properties, including the Euler–Poincaré principle (Theorem reference) and the exact sequence for \(\mathrm{K}_0\) of a Serre quotient (Proposition reference). These techniques are common in applications. The construction of \(\mathrm{K}_0\) applies to a broader class of structures than abelian categories, called exact categories; see [Bu10] for the theory. Moreover, \(\mathrm{K}_0\) is only the beginning of a sequence of abelian groups \(\mathrm{K}_n\) for \(n\in\ensuremath{\mathbb{Z}}_{\geq0}\). A satisfactory interpretation of the higher K-groups requires homotopy-theoretic ideas, especially the notion of a “spectrum”, and lies beyond the scope of this book.

Reading Guide. As noted above, readers should have some familiarity with concrete operations in module categories. For the basic theory of complexes and their cohomology, the material in §§reference—reference is essential. Apart from the treatment of abelian subcategories, the material in §reference is used less often later, but it remains standard knowledge and is related to several definitions in the discussion of derived categories. Section reference on Grothendieck categories is supplementary; it requires relatively involved arguments and material from the appendices, and may be skipped on a first reading.

A complex written as data \((X^n,d^n)_n\), where \(d^n:X^n\to X^{n+1}\), is also called a cochain complex. If instead one uses subscripts, writing \((X_n,d_n)_n\) with \(d_n:X_n\to X_{n-1}\), it is called a chain complex. Except in Chapter 8, this book primarily uses cochain notation.

Definition of an Abelian Category

Section reference introduced \(\ensuremath{\mathsf{Ab}}\)-categories and additive categories. The abelian categories studied in this chapter form a special class of additive categories.

Definition (Abelian category).

Let \(\mathcal{A}\) be an additive category. If every morphism in \(\mathcal{A}\) has a kernel and a cokernel, and every morphism is strict (Definition reference), then \(\mathcal{A}\) is called an abelian category.

If \(\mathcal{A}\) is also a \(\Bbbk\)-linear category (Definition reference), it is called a \(\Bbbk\)-linear abelian category.

Remark. An abelian category has all finite limits and colimits. Indeed, finite products and coproducts exist (both are biproducts), as do equalizers and coequalizers (Remark reference); these suffice to construct all finite limits and colimits. See [Li1].

By Example reference, \(R\ensuremath{\text{-}\mathsf{Mod}}\) is an abelian category for every ring \(R\). Replacing \(R\) by \(R^{\text{op}}\) shows the same for the category \(\ensuremath{\mathsf{Mod}\text{-}}R\) of right modules; the special case \(R=\ensuremath{\mathbb{Z}}\) shows that \(\ensuremath{\mathsf{Ab}}\) is abelian. By contrast, the category \(\ensuremath{\mathsf{Ban}}_{\ensuremath{\mathbb{C}}}\) from the same example is not abelian, since it has nonstrict morphisms.

Proposition. If an additive category \(\mathcal{A}\) is abelian, then so is \(\mathcal{A}^{\operatorname{op}}\).

Proof. Kernels and cokernels are dual, while Remark reference shows that the morphism \(\operatorname{coim}(f)\to\operatorname{im}(f)\) in \(\mathcal{A}\) is exactly the morphism \(\operatorname{coim}(f^{\operatorname{op}})\to\operatorname{im}(f^{\operatorname{op}})\) in \(\mathcal{A}^{\operatorname{op}}\).

One abstract way to construct new abelian categories from existing ones is to form functor categories. First note that if \(\mathcal{A}\) is an \(\ensuremath{\mathsf{Ab}}\)-category, then for every category \(I\)¹, the functor category \(\mathcal{A}^I\) is naturally an \(\ensuremath{\mathsf{Ab}}\)-category: for functors \(F,G:I\rightrightarrows\mathcal{A}\), addition in \(\operatorname{Hom}_{\mathcal{A}^I}(F,G)\) is defined objectwise by \((\varphi_i)_i+(\psi_i)_i:=(\varphi_i+\psi_i)_i\).

Proposition. If \(\mathcal{A}\) is an abelian category, then \(\mathcal{A}^I\) is an abelian category for every category \(I\).

Proof. By Proposition reference, finite products and coproducts, and kernels and cokernels, are constructed objectwise in \(\mathcal{A}^I\). The image and coimage of a morphism, and hence the canonical morphism \(\operatorname{coim}\to\operatorname{im}\), are likewise constructed objectwise. Thus every abelian category axiom for \(\mathcal{A}^I\) reduces to the corresponding axiom for \(\mathcal{A}\).

Every morphism \(f:X\to Y\) in an abelian category has a unique epi–mono factorization (Proposition reference), concretely \(X\twoheadrightarrow(\operatorname{coim}(f)\simeq\operatorname{im}(f))\hookrightarrow Y\). Lemma reference(iii) therefore gives immediately \[\operatorname{ker}(f) = \operatorname{ker}[X \to \operatorname{im}(f)], \quad \operatorname{coker}(f) = \operatorname{coker}[\operatorname{im}(f) \to Y].\]

For \(X'\xleftarrow{f'}Z'\xrightarrow{g'}Y'\), the fiber coproduct construction realizes \(X'\ensuremath{\underset{Z'}{\sqcup}}Y'\) as \(\operatorname{coker}[Z'\xrightarrow{(-f',g')}X'\oplus Y']\), and the morphisms \(X'\rightarrow X'\ensuremath{\underset{Z'}{\sqcup}}Y'\leftarrow Y'\) come from \(X'\xrightarrow{\iota_1}X'\oplus Y'\xleftarrow{\iota_2}Y'\). These properties are of course evident for modules.

The following result will be used repeatedly.

Proof. It suffices to treat the pullback case. We claim that

• \(X\xrightarrow{(a,f)}X'\oplus Y\) gives \(\operatorname{ker}[X'\oplus Y\xrightarrow{(g,-b)}Y']\);

• \((g,-b):X'\oplus Y\to Y'\) is epic.

The assertion about the kernel follows from the description of \(X\simeq X'\ensuremath{\underset{Y'}{\times}}Y\) as a kernel in Remark reference. Since \(g=(g,-b)\iota_1\), the fact that \(g\) is epic implies that \((g,-b)\) is epic as well.

Corollary. A subquotient in an abelian category can equivalently be defined as a quotient object of a subobject or as a subobject of a quotient object.

Proof. Proposition reference shows that Proposition reference applies to every abelian category.

For the category \(R\ensuremath{\text{-}\mathsf{Mod}}\) of left modules over any ring \(R\), all the results above are of course familiar; the same holds for right modules.

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1. If one does not want \(\mathcal{A}^I\) to be large, one should assume that \(I\) is small. This does not affect the substance here.↩︎

A First Look at Complexes

The starting point of this section is a pair of morphisms in an abelian category satisfying \(gf=0\), namely \(X \xrightarrow{f} Y \xrightarrow{g} Z\). The composite \(\operatorname{ker}(g) \hookrightarrow Y \twoheadrightarrow \operatorname{coker}(f)\) gives a morphism \(\operatorname{ker}(g) \to \operatorname{coker}(f)\); what interests us is its epi–mono factorization. There are two mutually dual points of view.

Lemma. Let \(f:X\to Y\) and \(g:Y\to Z\) be morphisms in an abelian category \(\mathcal{A}\) such that \(gf=0\).

1. There are unique morphisms \(\varphi\) and \(\psi\) that make the following diagrams commute: \[\begin{equation} \text{Diagram chapter2-unit-022-d001} \qquad \text{Diagram chapter2-unit-022-d002} \end{equation}\] Moreover, \(\varphi\) is a monomorphism and \(\psi\) is an epimorphism.

2. The morphism \(\operatorname{ker}(g)\to\operatorname{coker}(f)\) has the two epi–mono factorizations displayed in the following diagram; they are connected by a unique isomorphism \(\operatorname{coker}(\varphi)\simeq\operatorname{ker}(\psi)\): \[\text{Diagram chapter2-unit-022-d003}\]

Proof. First consider (i). The two diagrams are dual to one another (Remark reference and Proposition reference): the diagram on the right is equivalent to considering in \(\mathcal{A}^{\operatorname{op}}\) the sequence \(Z \xrightarrow{g^{\operatorname{op}}} Y \xrightarrow{f^{\operatorname{op}}} X\). Thus, for equation, it suffices to consider \(\varphi\).

The condition \(gf=0\) says precisely that \(f\) factors through \(\operatorname{ker}(g)\hookrightarrow Y\). Applying Lemma reference on coimages to this factorization gives \(\varphi\). Next, because \(X\to\operatorname{coim}(f)\) is an epimorphism, the factorization of \(f\) as \(X \twoheadrightarrow \operatorname{coim}(f) \xrightarrow{\overline{f}} Y\) is unique. Here \(\overline{f}\) may be taken either as the composite \(\operatorname{coim}(f) \xrightarrow{\varphi} \operatorname{ker}(g) \hookrightarrow Y\) or as the composite \(\operatorname{coim}(f) \ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}(f) \hookrightarrow Y\). The latter composite is a monomorphism, and hence so is \(\varphi\).

In the situation of Lemma reference, write

\[\begin{equation} \begin{aligned} \operatorname{H}\left[ X \xrightarrow{f} Y \xrightarrow{g} Z \right] & := \operatorname{coker}\left[ \operatorname{im}(f) \xrightarrow{\varphi} \operatorname{ker}(g) \right] \\ & \simeq \operatorname{ker}\left[ \operatorname{coker}(f) \xrightarrow{\psi} \operatorname{im}(g) \right] ; \end{aligned} \end{equation}\] this object can also be characterized as the middle term in the epi–mono factorization of \(\operatorname{ker}(g)\to\operatorname{coker}(f)\). As expected, equation is functorial.

Proposition. Given morphisms satisfying \(gf=0\) and \(g'f'=0\), there is a canonical isomorphism \[\begin{multline*} \operatorname{H}\left[ X \xrightarrow{f} Y \xrightarrow{g} Z \right] \oplus \operatorname{H}\left[ X' \xrightarrow{f'} Y' \xrightarrow{g'} Z' \right] \\ \simeq \operatorname{H}\left[ X \oplus X' \xrightarrow{f \oplus f'} Y \oplus Y' \xrightarrow{g \oplus g'} Z \oplus Z' \right]. \end{multline*}\]

The characterization of the induced morphism \(\Phi\) in Proposition reference also tells us that \((a,b,c)\mapsto\Phi\) is compatible with composition, preserves addition, sends \(0\) to \(0\), and sends \(\ensuremath{\mathrm{id}}\) to \(\ensuremath{\mathrm{id}}\). Hence the morphisms induced at the level of \(\operatorname{H}[\cdots]\) also satisfy the biproduct identities equation.

Notice that the preceding statement need not hold for infinite coproducts or infinite products in a general abelian category.

Definition (Complexes).

Let \(\mathcal{A}\) be an additive category. Consider a sequence of morphisms in \(\mathcal{A}\) \[\cdots \to X^n \xrightarrow{d^n} X^{n+1} \xrightarrow{d^{n+1}} X^{n+2} \to \cdots.\] The sequence may have finitely many terms, extend infinitely in one or both directions, or form a cycle. If \(d^{n+1}d^n=0\) for every \(n\), the data \((X^n,d^n)_n\) are called a complex; the customary notation is \((X^\bullet,d^\bullet)\), \(X^\bullet\), or \(X\). The term \(X^n\) is conventionally called the degree-\(n\) term of the complex \(X\).

This chapter is chiefly concerned with abelian categories, where the cohomology and exactness of complexes can be studied.

Definition (Cohomology and exact sequences).

Let \(\mathcal{A}\) be an abelian category.

• For a complex \(X\), in accordance with equation, its cohomology at any non-end term \(X^n\) is defined by \[\operatorname{H}^n(X) = \operatorname{H}^n(X^\bullet, d^\bullet) := \operatorname{H}\left[ X^{n-1} \xrightarrow{d^{n-1}} X^n \xrightarrow{d^n} X^{n+1} \right].\]

• If \(\operatorname{H}^n(X)=0\), the complex \(X\) is said to be exact at \(X^n\); this is equivalent to \(\operatorname{im}(d^{n-1})=\operatorname{ker}(d^n)\). A complex that is exact at every term is called an exact sequence; an exact complex is also called acyclic.

The properties of cohomology are one of the main themes of this book and will be studied further in §reference.

Remark (Homology of chain complexes).

In Definition reference, choose decreasing indices and use subscripts, namely \[\cdots \to X_n \xrightarrow{d_n} X_{n-1} \to \cdots, \quad d_{n-1} d_n = 0,\] and all the preceding definitions apply unchanged. By conventions originating in topology, the version with increasing indices \((X^\bullet,d^\bullet)\) is usually called a cochain complex, while the version with decreasing indices \((X_\bullet,d_\bullet)\) is called a chain complex. For a chain complex \(X\) in an abelian category, the homology object at \(X_n\) is defined as the following object of \(\mathcal{A}\): \[\operatorname{H}_n(X) = \operatorname{H}_n(X_\bullet, d_\bullet) := \operatorname{H}\left[ X_{n+1} \xrightarrow{d_{n+1}} X_n \xrightarrow{d_n} X_{n-1} \right].\]

In algebra, complexes (that is, cochain complexes) and chain complexes differ only in notation, and passing between them is straightforward. For an additive category \(\mathcal{A}\), one way to remain entirely within the same category is to reflect the indices by setting \[X^n := X_{-n}, \quad d^n := d_{-n}.\] Another way is to reverse the arrows, which involves the opposite category. If a complex in \(\mathcal{A}\) \[\cdots \to X^n \xrightarrow{d^n} X^{n+1} \to \cdots\] is viewed in \(\mathcal{A}^{\operatorname{op}}\), it becomes the chain complex \[\begin{gather*} \cdots \leftarrow X_n \xleftarrow{d_{n+1}} X_{n+1} \leftarrow \cdots, \\ X_n := X^n, \quad d_{n+1} := d^{n, \operatorname{op}}. \end{gather*}\]

If \(\mathcal{A}\) is an abelian category and the chain complex \((X_\bullet,d_\bullet)\) in \(\mathcal{A}^{\operatorname{op}}\) is defined by reversing the arrows, there is a canonical isomorphism \(\operatorname{H}^n(X^\bullet)\simeq\operatorname{H}_n(X_\bullet)\). In particular, \(X^\bullet\) is exact if and only if \(X_\bullet\) is exact. To see this, it suffices to note that \(\operatorname{H}[X \xrightarrow{f} Y \xrightarrow{g} Z]\) can be characterized as the middle term in the epi–mono factorization of \(\operatorname{ker}(g)\to\operatorname{coker}(f)\). Viewed in \(\mathcal{A}^{\operatorname{op}}\), the same object is also the middle term in the epi–mono factorization of \(\operatorname{ker}(f^{\operatorname{op}})\to\operatorname{coker}(g^{\operatorname{op}})\).

Exactness of a complex can be checked term by term. The following standard forms will henceforth be used without further explanation.

• Monomorphism.The complex \(0\to X'\xrightarrow{f}X\) is exact if and only if \(\operatorname{ker}(f)=0\), that is, if and only if \(f\) is a monomorphism (Lemma reference). This is also equivalent to \(X'\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{coim}(f)\) (Lemma reference).

• Epimorphism.Dually, the complex \(X\xrightarrow{g}X''\to0\) is exact if and only if \(g\) is an epimorphism, and this is also equivalent to \(\operatorname{im}(g)\ensuremath{\stackrel{\sim}{\rightarrow}}X''\).

• Kernel.Consider the complex \(0\to X'\xrightarrow{f}X\xrightarrow{g}X''\). In accordance with equation, factor \(f\) through \(X'\twoheadrightarrow\operatorname{im}(f)\hookrightarrow\operatorname{ker}(g)\). The claim is that \(0\to X'\to X\to X''\) is exact if and only if \(X'\to\operatorname{ker}(g)\) is an isomorphism. In other words, up to isomorphism, an exact sequence of this form is always \(0\to\operatorname{ker}(g)\to X\xrightarrow{g}X''\) and is given by the kernel of a morphism.

  The argument is easy. Consider equation. The complex is exact at \(X\) if and only if \(\operatorname{im}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{ker}(g)\), while exactness at \(X'\) is equivalent to \(X'\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}(f)\). Now apply Proposition reference to \(X'\twoheadrightarrow\operatorname{im}(f)\hookrightarrow\operatorname{ker}(g)\).

• Cokernel.Dually, the complex \(X'\xrightarrow{f}X\xrightarrow{g}X''\to0\) induces a morphism \(\operatorname{coker}(f)\to X''\) through equation. This complex is exact if and only if \(\operatorname{coker}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}X''\). Thus, up to isomorphism, an exact sequence of this form arises from a cokernel.

• Short exact sequence.An exact sequence of the form \(0\to X'\to X\to X''\to0\) is called a short exact sequence. The two preceding items together give two equivalent, mutually dual characterizations of exactness:

  • \(X'\to X\) is a monomorphism and \(X''=\operatorname{coker}[X'\hookrightarrow X]\);

  • \(X\to X''\) is an epimorphism and \(X'=\operatorname{ker}[X\twoheadrightarrow X'']\).

As an application, every morphism \(f:X\to Y\) gives short exact sequences \[0 \to \operatorname{ker}(f) \to X \to \operatorname{im}(f) \to 0, \quad 0 \to \operatorname{im}(f) \to Y \to \operatorname{coker}(f) \to 0,\] which can be spliced into the exact sequence \(0\to\operatorname{ker}(f)\to X\xrightarrow{f}Y\to\operatorname{coker}(f)\to0\). Next, suppose that \(f\) can be embedded in an exact sequence of the form \[L' \xrightarrow{\lambda} L \to X \xrightarrow{f} Y \to R \xrightarrow{\rho} R''\] with both \(\lambda\) and \(\rho\) isomorphisms. Exactness then forces both \(L\to X\) and \(Y\to R\) to be zero, so \(\operatorname{ker}(f)=0\) and \(\operatorname{im}(f)=Y\); in other words, \(f\) is an isomorphism. This is a commonly used technique for testing whether a morphism is an isomorphism. The biproducts considered in §reference give a very simple class of short exact sequences; Proposition reference will characterize them.

Proposition. Let \(X_1,X_2\) be objects of an abelian category \(\mathcal{A}\). Then \(0\to X_1\xrightarrow{\iota_1}X_1\oplus X_2\xrightarrow{p_2}X_2\to0\) is a short exact sequence.

Proof. Take the short exact sequences \(0\to X_1\xrightarrow{\ensuremath{\mathrm{id}}}X_1\to0\to0\) and \(0\to0\to X_2\xrightarrow{\ensuremath{\mathrm{id}}}X_2\to0\). Proposition reference shows that their direct sum is still exact.

Some Diagram Lemmas

The arguments in this section follow [KS06]. We begin with a criterion for exactness that may be viewed as a substitute for diagram chasing [Li1] in general abelian categories. Throughout this section, fix an abelian category \(\mathcal{A}\).

Proof. First consider the “only if” direction. The morphisms \(f\) and \(h\) both factor through \(\operatorname{ker}(g)\). On the basis of these factorizations, take \(S':=S\ensuremath{\underset{\operatorname{ker}(g)}{\times}}X'\), and take \(S'\to S\) and \(S'\to X'\) to be the natural morphisms from the fiber product. Exactness implies that \(X'\twoheadrightarrow\operatorname{ker}(g)\), so Proposition reference implies that \(S'\to S\) is also an epimorphism.

• \(X'\xrightarrow{f}X\xrightarrow{g}X''\to0\) and \(0\to Y'\xrightarrow{u}Y\xrightarrow{v}Y''\) are given exact sequences;

• \(\operatorname{ker}':=\operatorname{ker}[X'\to Y']\hookrightarrow X'\), with \(\operatorname{ker}\) and \(\operatorname{ker}''\) defined in the same way;

• \(Y'\twoheadrightarrow\operatorname{coker}':=\operatorname{coker}[X'\to Y']\), with \(\operatorname{coker}\) and \(\operatorname{coker}''\) defined in the same way;

• the morphisms \(\operatorname{ker}'\to\operatorname{ker}\), \(\operatorname{coker}'\to\operatorname{coker}\), and so forth come from the functoriality of kernels and cokernels in equation.

Thus equation is a commutative diagram; every column and both middle rows are exact. We now explain how to construct the dashed connecting morphism \(\delta:\operatorname{ker}''\to\operatorname{coker}'\).

• \(\Box\) and \(\boxplus\) indicate that the corresponding squares are, respectively, a pullback and a pushout diagram. Pullbacks preserve kernels, while pushouts preserve cokernels; this explains the upper-left and lower-right squares in the diagram;

• Proposition reference applies to the \(\Box\) and \(\boxplus\) parts of equation. This explains why \(X\ensuremath{\underset{X''}{\times}}\operatorname{ker}''\to\operatorname{ker}''\) is an epimorphism and \(\operatorname{coker}'\to\operatorname{coker}'\ensuremath{\underset{Y'}{\sqcup}}Y\) is a monomorphism;

• the dashed arrow \(\operatorname{ker}(g)\dashrightarrow Y'\) is none other than the natural morphism \(\operatorname{ker}(g)\to\operatorname{ker}(v)\) coming from the functoriality of kernels in equation;

• similarly, the functoriality of cokernels gives the dashed arrow \(X''\simeq\operatorname{coker}(f)\to\operatorname{coker}(u)\).

Notice that the composite \(\operatorname{ker}(g)\dashrightarrow Y'\to\operatorname{coker}'\) in equation is zero, because it becomes zero after precomposition with \(X'\twoheadrightarrow\operatorname{im}(f)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{ker}(g)\). Similarly, the composite \(\operatorname{ker}''\to X''\dashrightarrow\operatorname{coker}(u)\) is zero, because its further composite with \(\operatorname{coker}(u)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}(v)\hookrightarrow Y''\) is zero. Denote the composite along the middle path in equation by \(\delta_0:X\ensuremath{\underset{X''}{\times}}\operatorname{ker}''\to\operatorname{coker}'\ensuremath{\underset{Y'}{\sqcup}}Y\). The preceding observation immediately gives \[\delta_0 s = 0, \quad t \delta_0 = 0.\] Consequently, \(\delta_0\) has a unique factorization \[X \ensuremath{\underset{X''}{\times}} \operatorname{ker}'' \twoheadrightarrow \operatorname{coker}(s) \xrightarrow{\exists!} \operatorname{ker}(t) \hookrightarrow \operatorname{coker}' \ensuremath{\underset{Y'}{\sqcup}} Y .\]

The horizontal morphisms in the upper-right and lower-left corners of equation are known to be an epimorphism and a monomorphism, respectively. Hence \(\operatorname{coker}(s)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{ker}''\) and \(\operatorname{coker}'\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{ker}(t)\). We have therefore obtained the desired canonical morphism \(\delta:\operatorname{ker}''\to\operatorname{coker}'\).

The following two observations will be used in the proof below.

• The construction of the connecting morphism is self-dual: when equation is viewed in \(\mathcal{A}^{\operatorname{op}}\), all arrows are reversed, while the roles of \(\operatorname{ker}'\) and \(\operatorname{coker}'\), and so forth, are interchanged. The resulting diagram in \(\mathcal{A}^{\operatorname{op}}\) still resembles equation, except that it is rotated through half a turn. When the corresponding connecting morphism is viewed back in \(\mathcal{A}\), it still runs from \(\operatorname{ker}''\) to \(\operatorname{coker}'\), and it agrees with the \(\delta\) constructed above.

• The morphism \(\operatorname{ker}\to\operatorname{ker}''\) factors as \(\operatorname{ker}\to X\ensuremath{\underset{X''}{\times}}\operatorname{ker}''\to\operatorname{ker}''\). Inspection of equation shows that the two composites \[\operatorname{ker}\to \operatorname{ker}'' \xrightarrow{\delta} \operatorname{coker}' \hookrightarrow \operatorname{coker}' \ensuremath{\underset{Y'}{\sqcup}} Y, \qquad \operatorname{ker}\to X \ensuremath{\underset{X''}{\times}} \operatorname{ker}'' \xrightarrow{\delta_0} \operatorname{coker}' \ensuremath{\underset{Y'}{\sqcup}} Y\] are equal. But the second composite is also equal to \(\operatorname{ker}\to X\to Y\to\operatorname{coker}'\ensuremath{\underset{Y'}{\sqcup}}Y\), which is zero. We therefore obtain \[\begin{equation} \operatorname{ker}\to \operatorname{ker}'' \xrightarrow{\delta} \operatorname{coker}' \quad \text{has composite}\; 0. \end{equation}\]

Theorem (Snake Lemma). Consider diagram equation in an abelian category \(\mathcal{A}\). Then \(\operatorname{ker}'\to\operatorname{ker}\to\operatorname{ker}''\xrightarrow{\delta}\operatorname{coker}'\to\operatorname{coker}\to\operatorname{coker}''\) is an exact sequence. If \(f:X'\to X\) in the diagram is a monomorphism (or \(v:Y\to Y''\) is an epimorphism), then \(\operatorname{ker}'\to\operatorname{ker}\) is also a monomorphism (or \(\operatorname{coker}\to\operatorname{coker}''\) is also an epimorphism).

Proof. The last assertion is easy to prove: if \(f\) is a monomorphism, then the composite \(\operatorname{ker}'\hookrightarrow X'\xrightarrow{f}X\) is also a monomorphism. Since equation commutes, this implies that \(\operatorname{ker}'\to\operatorname{ker}\) is a monomorphism. The case in which \(v\) is an epimorphism is obtained by reversing all arrows.

The main part of the proof is the exactness of the first sequence. By the duality observed above, it suffices to prove that \(\operatorname{ker}'\to\operatorname{ker}\to\operatorname{ker}''\to\operatorname{coker}'\) is exact.

Now consider the left part of equation: the outer frame and the lower square commute. The familiar Lemma reference shows that the upper square also commutes. Therefore \(\operatorname{ker}'\to\operatorname{ker}\to\operatorname{ker}''\) is exact.

The composite along the first row is \(\delta\psi=0\), so the composite along the second row is also zero. Consequently, the composite \(S_1\xrightarrow{\xi}Y'\xrightarrow{u}Y\to\operatorname{coker}'\ensuremath{\underset{Y'}{\sqcup}}Y\) is zero. On the other hand, the composite \(Y'\xrightarrow{u}Y\to\operatorname{coker}'\ensuremath{\underset{Y'}{\sqcup}}Y\) equals \(Y'\to\operatorname{coker}'\to\operatorname{coker}'\ensuremath{\underset{Y'}{\sqcup}}Y\). In the construction of \(\delta\) we proved that \(\operatorname{coker}'\to\operatorname{coker}'\ensuremath{\underset{Y'}{\sqcup}}Y\) is a monomorphism. Thus the composite \(S_1\xrightarrow{\xi}Y'\to\operatorname{coker}'\) is indeed zero.

The connecting homomorphism \(\delta\) and Theorem reference are much simpler in the category \(R\ensuremath{\text{-}\mathsf{Mod}}\) of \(R\)-modules; see [Li1].

Proposition (Five Lemma).

1. If \(f_1\) is an epimorphism and \(f_2,f_4\) are monomorphisms, then \(f_3\) is a monomorphism (only the first four columns are involved);

2. if \(f_5\) is a monomorphism and \(f_2,f_4\) are epimorphisms, then \(f_3\) is an epimorphism (only the last four columns are involved);

3. if \(f_1\) is an epimorphism, \(f_5\) is a monomorphism, and \(f_2,f_4\) are both isomorphisms, then \(f_3\) is an isomorphism.

Proof. Clearly (i) and (ii) are dual to each other, while (iii) follows from (i)–(ii) and Proposition reference. It therefore suffices to prove (i).

Thus the composite \(S'''\twoheadrightarrow S'\twoheadrightarrow S\xrightarrow{h}X_3\) is zero, and hence \(h=0\). This proves the result.

This section has introduced only two of the most commonly used diagram lemmas. Using the language of double complexes, G. Bergman found the comprehensive Salamander Lemma, which subsumes the Snake Lemma and several classical diagram lemmas of homological algebra. Interested readers may consult [Be12].

A Glimpse of Lattice Theory

This section gives a brief introduction to a class of partially ordered structures called lattices. The discussion will help clarify the structure of abelian categories. Throughout this section we consider various partially ordered sets \((P,\leq)\); the corresponding category is denoted by \(\mathcal{P}\). For every subset \(S\) of a partially ordered set \((P,\leq)\), there are notions of upper and lower bounds, a supremum \(\sup S\), and an infimum \(\inf S\). A subset \(S\subset P\) has a supremum (or infimum) if and only if the coproduct (or product) of all the objects in \(S\) exists in the category \(\mathcal{P}\); in that case the supremum (or infimum) is precisely that coproduct (or product).

Definition.

If every two elements \(a,b\) of a partially ordered set \((P,\leq)\) have a supremum \(a\vee b\) and an infimum \(a\wedge b\), then \((P,\leq)\) is called a lattice. If, in addition, \((P,\leq)\) itself has an upper bound and a lower bound, they are both unique and are denoted by \(1\) and \(0\), respectively; in this case \(P\) is called a bounded lattice.

If \(a,b\) are elements of a partially ordered set \((P,\leq)\) and \(a\leq b\), then \([a,b]:=\left\{x\in P:a\leq x\leq b\right\}\) is also a partially ordered set under \(\leq\) and is called the interval determined by \(a,b\).

If \(P\) is a lattice, every interval in \(P\) is closed under \(\vee\) and \(\wedge\) and is a bounded lattice. If \(P\) is a bounded lattice, then \(P=[0,1]\), and for every \(x\in P\) one has \(x\wedge0=0\), \(x\vee0=x=x\wedge1\), and \(x\vee1=1\). From the categorical point of view, \(a\wedge b\), \(a\vee b\), \(1\), and \(0\) correspond, respectively, to the product of two objects, the coproduct of two objects, a terminal object (that is, an empty product), and an initial object (that is, an empty coproduct) in the category; this can be checked directly. As an illustration of the operations \(\wedge\) and \(\vee\), prove the following fact. If \(a^\flat,a,b\in P\) satisfy \(a^\flat\leq a\), then \[\begin{equation} a^\flat \vee (a \wedge b) \leq (a^\flat \vee b) \wedge a. \end{equation}\] By the definition of a supremum, it first suffices to prove \(a^\flat\leq(a^\flat\vee b)\wedge a\) and \((a\wedge b)\leq(a^\flat\vee b)\wedge a\). By the definition of an infimum, it then suffices to prove, respectively, the inequalities \[a^\flat \leq a^\flat \vee b, \quad a^\flat \leq a, \quad a \wedge b \leq a^\flat \vee b, \quad a \wedge b \leq a;\] the third follows from \(a\wedge b\leq b\leq a^\flat\vee b\), and the others are clear.

Definition.

Let \(P\) be a lattice.

• \(P\) is called a modular lattice if the following property holds: whenever \(a^\flat,a,b\in P\) satisfy \(a^\flat\leq a\), then \[a^\flat \vee (a \wedge b) = (a^\flat \vee b) \wedge a.\]

• Suppose that \(P\) is a bounded lattice. If \(x,c\in P\) satisfy \(x\vee c=1\) and \(x\wedge c=0\), then \(c\) is called a complement of \(x\) in \(P\).

We now consider some basic examples of lattices.

• The set of positive integers \(\ensuremath{\mathbb{Z}}_{\geq1}\) is a lattice under divisibility (\(x\mid y\iff x\leq y\)): \(x\vee y=\mathrm{lcm}(x,y)\) and \(x\wedge y=\mathrm{gcd}(x,y)\). The reader can check that this lattice is modular, has lower bound \(1\), but has no upper bound.

• Let \(R\) be a ring and \(M\) a left \(R\)-module. Its set of submodules \(\mathrm{Sub}_M\) is a bounded modular lattice under \(\subset\): \(x\vee y=x+y\) and \(x\wedge y=x\cap y\); its upper bound is \(M\) and its lower bound is \(\{0\}\). This is the origin of the term “modular lattice.”

• All closed subspaces of a Hilbert space \(H\) form a bounded lattice under \(\subset\): \(x\vee y:=\overline{x+y}\) and \(x\wedge y=x\cap y\). One can prove that this lattice is not modular when \(H\) is infinite-dimensional (Birkhoff–von Neumann).

Remark. The definition of a lattice also contains a duality. Given the partial order \(\leq\) on a set \(P\), we may consider the opposite partial order \(\leq^{\operatorname{op}}\): \(x\leq^{\operatorname{op}}y\iff x\geq y\); this is equivalent to replacing \(\mathcal{P}\) by \(\mathcal{P}^{\operatorname{op}}\). If \((P,\leq)\) is a lattice (or a bounded lattice), then so is \((P,\leq^{\operatorname{op}})\). Moreover, \(\wedge,\vee,0,1\) in \((P,\leq)\) correspond, respectively, to \(\vee,\wedge,1,0\) in \((P,\leq^{\operatorname{op}})\).

Lemma. Let \(P\) be a lattice. Then \(P\) is modular if and only if the following property holds for every interval \(I\) in \(P\): if \(c^\flat,c\) are both complements of \(x\in I\) in \(I\) and \(c^\flat\leq c\), then \(c^\flat=c\).

Proof. If \(P\) is a modular lattice, then every interval \(I\) is also a modular lattice. Without loss of generality, assume \(I=P\); for the \(x,c^\flat,c\) in the statement, we have \[c = c \wedge 1 = c \wedge (x \vee c^\flat) = c^\flat \vee (x \wedge c) = c^\flat \vee 0 = c^\flat .\]

Conversely, assume that the condition on complements holds for every interval. Let \(a^\flat,a,b\in P\) satisfy \(a^\flat\leq a\). Set \[b \wedge a \leq c_1 := a^\flat \vee (b \wedge a) \underset{\because \href{#eqn:modular-inequality}{equation}}{\leq} a \wedge (b \vee a^\flat) =: c_2 \leq b \vee a^\flat.\] We prove that \(c_1,c_2\) are both complements of \(b\) in \(\left[b\wedge a,\;b\vee a^\flat\right]\), so that \(c_1=c_2\). First, the definition immediately gives \(c_1\leq a\); therefore \[b \wedge a \geq b \wedge c_1 = (a^\flat \vee (b \wedge a)) \wedge b \underset{\because \href{#eqn:modular-inequality}{equation}}{\geq} (a^\flat \wedge b) \vee (b \wedge a) = b \wedge a ,\] so \(b\wedge c_1=b\wedge a\). On the other hand, \(b\vee c_1=a^\flat\vee(b\wedge a)\vee b=a^\flat\vee b\). Thus \(c_1\) is indeed a complement of \(b\) in \(\left[b\wedge a,\;b\vee a^\flat\right]\).

Lattice duality (Remark reference) reverses the order relation between \(a^\flat,a\), interchanges \(\wedge\) and \(\vee\), and interchanges the roles of \(c_1\) and \(c_2\). Thus the preceding argument, carried out in \((P,\leq^{\operatorname{op}})\), shows that \(c_2\) is a complement of \(b\) in \(\left[b\wedge a,\;b\vee a^\flat\right]\).

Proposition.

Proof. The two maps are clearly well defined and order-preserving. For \(x\in[a\wedge b,a]\), the definition of a modular lattice gives \((x\vee b)\wedge a=(a\wedge b)\vee x\), whose right-hand side is \(x\). Dually, \(y\in[b,a\vee b]\) implies \((y\wedge a)\vee b=y\) (Remark reference); thus the two maps are mutually inverse.

The Zassenhaus lemma of elementary algebra (see [Li1], especially its module-theoretic version) and its consequences can be distilled using lattice theory.

Theorem (Zassenhaus Lemma for Modular Lattices).

Proof. The isomorphisms in the last part are obtained by applying Proposition reference to the two parallelograms in the diagram; what remains to be checked are the relations in equation. Notice that the diagram is left–right symmetric under \(u\leftrightarrow v\) and \(u^\flat\leftrightarrow v^\flat\); it therefore suffices to check its left half. The partial-order relations among its elements are immediate. For the \(\wedge\) case, the definition of a modular lattice gives \[\begin{equation*} \left( u^\flat \vee (u \wedge v^\flat)\right) \wedge (u \wedge v) = \left( u^\flat \wedge (u \wedge v) \right) \vee (u \wedge v^\flat) = (u^\flat \wedge v) \vee (u \wedge v^\flat). \end{equation*}\] For the \(\vee\) case, the inequality \(u\wedge v^\flat\leq u\wedge v\) immediately gives \[\begin{equation*} u^\flat \vee (u \wedge v^\flat) \vee (u \wedge v) = u^\flat \vee (u \wedge v). \end{equation*}\] This proves equation.

Next consider a finite descending chain in a lattice, written \(x_0\geq x_1\geq\cdots\geq x_r\) (\(r\in\ensuremath{\mathbb{Z}}_{\geq0}\)). A descending chain obtained by inserting finitely many intermediate elements is called a refinement of the original descending chain; if the inserted elements include some \(y\notin\{x_0,\ldots,x_r\}\), it is called a proper refinement.

Definition. Two descending chains of the same length in a modular lattice, \(x_0\geq\cdots\geq x_r\) and \(x'_0\geq\cdots\geq x'_r\), are called equivalent if the following condition holds: there is a bijection \(\sigma\) from \(\{0,\ldots,r-1\}\) to itself (that is, a permutation)¹ such that, for every \(0\leq i<r\), there is an isomorphism of partially ordered sets \[\tau_i: \left[ x_{i+1}, x_i \right] \simeq \left[ x'_{\sigma(i) + 1}, x'_{\sigma(i)} \right],\] and \(\tau_i\) decomposes into a finite composite of standard isomorphisms (Proposition reference) or their inverses.

Theorem (Schreier Refinement Theorem for Modular Lattices).

Consider descending chains \[x_0 \geq \cdots \geq x_r, \quad y_0 \geq \cdots \geq y_s,\] in a modular lattice \((P,\leq)\) such that \(x_0=y_0\) and \(x_r=y_s\), with \(r,s\in\ensuremath{\mathbb{Z}}_{\geq0}\). Then each chain has a refinement, and the two refinements are equivalent.

Proof. The argument is the same as in the case of groups [Li1]; both are based on Theorem reference. We give the outline. Define \[x_{i, j} := x_{i+1} \vee (x_i \wedge y_j), \quad y_{j, i} := (x_i \wedge y_j) \vee y_{j+1},\] with \(0\leq i<r\) and \(0\leq j\leq s\) for \(x_{i,j}\), and with \(0\leq i\leq r\) and \(0\leq j<s\) for \(y_{j,i}\). Then \(x_{i,j+1}\leq x_{i,j}\), \(x_{i,0}=x_i\), and \(x_{i,s}=x_{i+1}\), so \(\left(x_{i,j}\right)_{i,j}\), in that order, refines \(x_0\geq\cdots\geq x_r\). Similarly, \(\left(y_{j,i}\right)_{j,i}\) refines \(y_0\geq\cdots\geq y_s\). Taking \(u^\flat=x_{i+1}\), \(u=x_i\), \(v^\flat=y_{j+1}\), and \(v=y_j\) in Theorem reference, we obtain \[\left[ x_{i, j+1}, x_{i,j} \right] \simeq \left[ y_{j, i+1}, y_{j, i} \right],\] and this isomorphism decomposes into standard isomorphisms and their inverses. The assertion follows.

From now on, we consider only strictly ascending or strictly descending chains.

Definition. Fix a partially ordered set \(P\) and elements \(a\leq b\) in it.² If a descending chain \(b=x_0>\cdots>x_r=a\) in \(P\) has no proper refinement, it is called a composition series of \([a,b]\).

The length of this composition series is defined to be \(r\in\ensuremath{\mathbb{Z}}_{\geq0}\); notice that \(r=0\) if and only if \(a=b\).

Theorem (Jordan–Hölder Theorem for Modular Lattices).

Let \(a<b\) be elements of a modular lattice \((P,\leq)\). Then any two composition series of \([a,b]\) have the same length and are equivalent in the sense of Definition reference.

Proof. This follows immediately from Theorem reference.

The next important question is to determine which intervals in a partially ordered set have composition series. As in the familiar theory of modules, their existence can be guaranteed by ascending- and descending-chain conditions.

Definition.

Let \((P,\leq)\) be a partially ordered set. If there is no infinite ascending chain \(x_1<x_2<x_3<\cdots\) in \(P\), then \(P\) is called Noetherian; if there is no infinite descending chain \(x_1>x_2>x_3>\cdots\), then \(P\) is called Artinian. A partially ordered set that is both Artinian and Noetherian is said to have finite length.

If \((P,\leq)\) is Noetherian (or Artinian, or of finite length), then every subset of it has the same property. It is easy to prove that the Noetherian (or Artinian) condition is equivalent to the assertion that every nonempty subset of \(P\) has a maximal (or minimal) element with respect to \(\leq\). In this chapter these notions will be applied chiefly to partially ordered sets of subobjects; see Definition reference.

Definition.

Fix a category \(\mathcal{C}\). An object \(X\) of \(\mathcal{C}\) is called Noetherian (or Artinian, or of finite length) if its partially ordered set of subobjects \((\mathrm{Sub}_X,\subset)\) is Noetherian (or Artinian, or of finite length).

We now return to general partially ordered sets.

Lemma. Let \(a\leq b\) be elements of a partially ordered set \((P,\leq)\).

1. If \([a,b]\) has finite length, then every chain in it can be refined to a composition series; in particular, \([a,b]\) has a composition series.

2. If \([a,b]\) is a modular lattice and has a composition series, then \([a,b]\) has finite length.

Proof. Without loss of generality, assume \(a<b\). For (i), given a chain \(y_0>\cdots>y_r\), it suffices to show that every \([y_{i+1},y_i]\) has a composition series.³ Without loss of generality, assume \(y_{i+1}=a\) and \(y_i=b\). The Artinian condition guarantees a minimal element \(x^0\in[a,b]\) with \(x^0>a\). Similarly, if \(x^0\neq b\), there is a minimal element \(x^1\in[a,b]\) with \(x^1>x^0\), and so on. By the Noetherian condition, this process must stop after finitely many steps, producing a chain \(b>\cdots>x^0>a\) with no proper refinement.

For (ii), choose a composition series \(b=x_0>\cdots>x_r=a\). Theorem reference shows that every finite chain \(\cdots>y_i>y_{i+1}>\cdots\) has a refinement equivalent to the composition series above; hence the length of that chain cannot exceed the length of the composition series. Thus \([a,b]\) is a partially ordered set of finite length.

Definition. Let \((P,\leq)\) be a bounded modular lattice of finite length. Choose any composition series \(1=x_0>\cdots>x_r=0\) of \(P\). The number \(r\in\ensuremath{\mathbb{Z}}_{\geq0}\) is called the length of \((P,\leq)\).

Theorem reference shows that this length is well defined and does not depend on the chosen composition series; the length of \((P,\leq)\) is \(0\) if and only if \(P\) is a one-point set.

---

1. Translator’s note (O014-C021): the source has \(\{0,\ldots,r\}\), but the successive factors being paired are indexed by \(0\leq i<r\); the index \(i=r\) would leave \(x_{i+1}\) undefined.↩︎

2. Translator’s note (O014-C022): the source has \(a<b\), but the next sentence explicitly includes the case \(a=b\) and \(r=0\); the target uses the weaker hypothesis consistent with that case.↩︎

3. Translator’s note (O014-C023): the source has \([y_i,y_{i+1}]\) and then sets \(y_i=a\), \(y_{i+1}=b\). Since \(y_i>y_{i+1}\), both orders are reversed; the target uses the interval \([y_{i+1},y_i]\) and the well-typed assignment.↩︎

Direct-Sum Decompositions

We begin by discussing how morphisms between direct sums are represented by matrices. The argument is exactly the same as in the module-theoretic case [Li1]; only a brief review is given here. Fix an additive category \(\mathcal{A}\). Consider objects \(X_1,\ldots,X_n\) and \(X'_1,\ldots,X'_m\) of \(\mathcal{A}\). The direct sum \(\bigoplus_{i=1}^nX_i\) introduced in Definition reference comes with a family of morphisms \(X_j\xrightarrow{\iota_j}\bigoplus_{i=1}^nX_i\xrightarrow{p_j}X_j\). Likewise, \(\bigoplus_{i=1}^mX'_i\) has morphisms \(p'_j,\iota'_j\), and so on. The universal properties of products and coproducts give \[\begin{align*} \operatorname{Hom}\left( \bigoplus_{j=1}^n X_j, \bigoplus_{i=1}^m X'_i \right) & \xrightarrow[\sim]{\phi \mapsto (p'_i \phi)_i } \prod_{i=1}^m \operatorname{Hom}\left(\bigoplus_{j=1}^n X_j , X'_i \right) \\ & \xrightarrow[\sim]{(p'_i \phi)_i \mapsto (p'_i \phi \iota_j)_{i, j}} \prod_{j=1}^n \prod_{i=1}^m \operatorname{Hom}(X_j, X'_i). \end{align*}\] Thus every morphism \(\phi:\bigoplus_{j=1}^nX_j\to\bigoplus_{i=1}^mX'_i\) corresponds to the matrix \[\mathcal{M}(\phi) = (\phi_{ij})_{\substack{1 \leq i \leq m \\ 1 \leq j \leq n}} = \begin{pmatrix} \phi_{11} & \cdots & \phi_{1n} \\ \vdots & \ddots & \vdots \\ \phi_{m1} & \cdots & \phi_{mn} \end{pmatrix}, \quad \phi_{ij} := p'_i \phi \iota_j: X_j \to X'_i.\] Conversely, every matrix \(\mathcal{M}=(\phi_{ij})_{\substack{1\leq i\leq m\\1\leq j\leq n}}\) formed from morphisms \(\phi_{ij}:X_j\to X'_i\) uniquely determines a morphism \(\phi\) such that \(\mathcal{M}=\mathcal{M}(\phi)\). Composition of morphisms agrees with matrix multiplication: \[\mathcal{M}(\psi\phi) = \mathcal{M}(\psi) \mathcal{M}(\phi)\] where multiplication of matrix entries is given by composition \(\operatorname{Hom}(X'_j,X''_i)\times\operatorname{Hom}(X_k,X'_j)\to\operatorname{Hom}(X_k,X''_i)\).

Lemma. Given a family of morphisms \(\phi_i:X_i\to X'_i\) in \(\mathcal{A}\), \(i=1,\ldots,n\), set \(\phi:=(\phi_1,\ldots,\phi_n):\bigoplus_{i=1}^nX_i\to \bigoplus_{i=1}^nX'_i\). Then \(\phi\) is an isomorphism if and only if each \(\phi_i\) is an isomorphism.

Proof. The implication \(\impliedby\) is clear. For \(\implies\), write write the matrix \(\mathcal{M}(\phi^{-1})\) as \((\psi_{ij})_{i,j}\). Then \[\mathcal{M}(\phi^{-1}) \mathcal{M}(\phi) = \begin{pmatrix} \psi_{11} \phi_1 & \cdots & \psi_{1n} \phi_n \\ \vdots & \ddots & \vdots \\ \psi_{n1} \phi_1 & \cdots & \psi_{nn} \phi_n \end{pmatrix} = \begin{pmatrix} \ensuremath{\mathrm{id}}_{X_1} & & \\ & \ddots & \\ & & \ensuremath{\mathrm{id}}_{X_n} \end{pmatrix}\] so \(\phi_1,\ldots,\phi_n\) all have left inverses. The same argument shows that they all have right inverses.

Continue to consider a direct-sum decomposition \(X\simeq\bigoplus_{i=1}^nX_i\), where \(n\geq1\) and \(X_i\neq0\) for every \(i\). It gives morphisms \(\iota_i:X_i\to X\) and \(p_i:X\to X_i\). For each \(1\leq i\leq n\), set \(e_i:=\iota_ip_i\in\operatorname{End}(X)\). These morphisms have the following properties. Recall that an idempotent in the ring \(\operatorname{End}(X)\) is an element \(e\in\operatorname{End}(X)\) satisfying \(e^2=e\); when no confusion can arise, we also call \(e\) an idempotent in \(\mathcal{A}\).

1. Every \(e_i\) is idempotent: \(e_i^2=(\iota_ip_i)(\iota_ip_i)=\iota_i(p_i\iota_i)p_i=\iota_ip_i\).

2. Orthogonality: \(i\neq j\implies e_ie_j=\iota_ip_i\iota_jp_j=0\).

3. The equality \(\sum_{i=1}^ne_i=1\) holds in the ring \(\operatorname{End}(X)\).

From now on we regard the direct summands \(X_i\) as subobjects of \(X\) and write the decomposition as an equality \(X=\bigoplus_{i=1}^nX_i\). Conversely, if \(\mathcal{A}\) is abelian, a family of idempotents \(e_1,\ldots,e_n\in\operatorname{End}(X)\) satisfying (E1)–(E3) determines subobjects of \(X\) by \[X_i := \operatorname{ker}\left( \sum_{j \neq i} e_j \right) = \operatorname{im}(e_i), \quad i=1, \ldots, n.\] On the one hand they come with monomorphisms \(X_i\xrightarrow{\iota_i}X\); on the other, \(e_i:X\to X\) factors uniquely as \(X\xrightarrow{p_i}X_i\xrightarrow{\iota_i}X\). This is precisely the data required for a direct sum. Passing from the idempotents \(e_1,\ldots,e_n\) to the direct summands \(X_1,\ldots,X_n\) does not require all the properties of an abelian category. It is enough to assume that \(\mathcal{A}\) is an \(\ensuremath{\mathsf{Ab}}\)-category with a zero object in which every idempotent has a kernel. Such a category is called a Karoubian category or a pseudo-abelian category. The exercises of this chapter discuss this further.

Proof. See [Li1].

In the special case \(n=2\), direct-sum decompositions are related to a special class of short exact sequences, called split short exact sequences.

Proposition.

For a short exact sequence \(0\to X'\xrightarrow{f}X\xrightarrow{g}X''\to0\) in an abelian category, the following statements are equivalent.

1. There is an \(s:X''\to X\) such that \(gs=\ensuremath{\mathrm{id}}_{X''}\).

2. There is an \(r:X\to X'\) such that \(rf=\ensuremath{\mathrm{id}}_{X'}\).

3. There is a diagram \(\text{Diagram chapter2-unit-025-diagram-002}\) for which \(X\simeq X'\oplus X''\); see the review of biproducts in §reference.

4. The map \(g_*:\operatorname{Hom}(S,X)\to\operatorname{Hom}(S,X'')\), sending \(\varphi\) to \(g\varphi\), is surjective for every object \(S\).

5. The map \(f^*:\operatorname{Hom}(X,S)\to\operatorname{Hom}(X',S)\), sending \(\psi\) to \(\psi f\), is surjective for every object \(S\).

If any of these conditions holds, the short exact sequence \(0\to X'\to X\to X''\to0\) is called split. The morphism \(s:X''\to X\) in (i), the morphism \(r:X\to X'\) in (ii), ¹ and the isomorphism \(X\ensuremath{\stackrel{\sim}{\rightarrow}}X'\oplus X''\) in (iii), denoted by \(\Phi\), correspond as follows:

• from \(\Phi\) to \(s\), take the composite \(X''\xrightarrow{\iota_2}X'\oplus X''\xrightarrow{\Phi^{-1}}X\);

• from \(s\) to \(r\), factor \(\ensuremath{\mathrm{id}}_X-sg:X\to X\) uniquely as \(X\xrightarrow{r}X'\xrightarrow{f}X\); this is the desired \(r\);

• from \(r\) to \(\Phi\), take \((r,g):X\to X'\oplus X''\), which is an isomorphism.

Moreover, for a given \(s\) or its corresponding \(r\), the orthogonal idempotents corresponding to the direct-sum decomposition in (iii) are \[e' := fr = \ensuremath{\mathrm{id}}_X - sg, \quad e'' := sg = \ensuremath{\mathrm{id}}_X - fr.\]

Proof. For modules, see [Li1]. Its proof uses only algebraic operations in \(\operatorname{Hom}\) sets and the characterization of biproducts, all of which also hold in an abelian category, so the argument carries over without change. We omit the details.

In the setting of Proposition reference, given \(f:X'\to X\) and \(g:X\to X''\), an \(s\) satisfying \(gs=\ensuremath{\mathrm{id}}_{X''}\) is called a section of \(g\), while an \(r\) satisfying \(rf=\ensuremath{\mathrm{id}}_{X'}\) is called a retraction of \(f\); these terms come from topology. The existence of a section or retraction implies, respectively, that \(g\) is an epimorphism or \(f\) is a monomorphism, but the converses do not hold.

Notice also that the five statements (i)–(v) of the proposition are collectively self-dual.

Definition. Let \(S\) be a nonzero object of an abelian category \(\mathcal{A}\). If \(S=S'\oplus S''\) implies \(S'=0\) or \(S''=0\), then \(S\) is called indecomposable.

Corollary. A nonzero object \(X\) of an abelian category is indecomposable if and only if \[\forall e \in \operatorname{End}(X), \quad e^2 = e \iff (e = 0 \;\vee\; e = 1).\]

Proof. Apply Proposition reference.

We wish to study decompositions \(X=X_1\oplus\cdots\oplus X_n\), where \(X\neq0\) and every \(X_i\) is indecomposable. There are two questions: existence and uniqueness. For modules this is the content of the Krull–Remak–Schmidt theorem; see [Li1]. For a general abelian category, some preparation is needed. We follow the approach of [Kr15].

Definition.

The following result generalizes [Li1].

Lemma (Fitting’s lemma in an abelian category). Let \(\mathcal{A}\) be an abelian category, let \(X\) be a nonzero object satisfying the bi-chain condition, and let \(f\in\operatorname{End}(X)\).

1. For \(n\gg0\), \(\operatorname{ker}(f^n)\) and \(\operatorname{im}(f^n)\) are independent of \(n\); denote them by \(\operatorname{ker}(f^\infty)\) and \(\operatorname{im}(f^\infty)\), respectively. Then \(X=\operatorname{ker}(f^\infty)\oplus\operatorname{im}(f^\infty)\).

2. If \(X\) is indecomposable, then \(f\) is either invertible or nilpotent in the ring \(\operatorname{End}(X)\).

Proof. For \(n\in\ensuremath{\mathbb{Z}}_{\geq0}\), define \(X_n:=\operatorname{im}(f^n)\), so \(X_0=X\). Proposition reference and \(\operatorname{coim}\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}\) give an epimorphism \(\alpha_n:X_n\to X_{n+1}\) induced by \(f\) and a monomorphism \(\beta_n:X_{n+1}\to X_n\). Thus we obtain a bi-chain \((X_n,\alpha_n,\beta_n)_{n=0}^\infty\).

For \(n\gg0\), \(\alpha_n\) and \(\beta_n\) are isomorphisms. Hence the subobject \(\operatorname{im}(f^\infty):=\operatorname{im}(f^n)\) of \(X\), for \(n\gg0\), is well defined; write its monomorphism as \(\iota:\operatorname{im}(f^\infty)\hookrightarrow X\). Likewise, \(\operatorname{ker}(f^n)=\operatorname{ker}[X\to\operatorname{im}(f^n)]\) is a subobject \(\operatorname{ker}(f^\infty)\) independent of \(n\) for \(n\gg0\).

For \(n\gg0\), the morphism \(\alpha_{2n-1}\cdots\alpha_n:X_n\to X_{2n}\) is invertible. Denote its inverse by \(\psi:\operatorname{im}(f^\infty)\ensuremath{\stackrel{\sim}{\rightarrow}}\operatorname{im}(f^\infty)\) and set \[p := \psi f^n: X \to \operatorname{im}(f^\infty), \quad \operatorname{ker}(p) = \operatorname{ker}(f^\infty) \qquad (n \gg 0).\] Since \(p\iota=\ensuremath{\mathrm{id}}_{\operatorname{im}(f^\infty)}\), the endomorphism \(e:=\iota p\in\operatorname{End}(X)\) is idempotent, with \(\operatorname{im}(e)=\operatorname{im}(f^\infty)\) and \(\operatorname{ker}(e)=\operatorname{ker}(f^\infty)\). Proposition reference gives the decomposition in (i).

If \(X\) is indecomposable, then either \(\operatorname{im}(f^\infty)=0\) and \(\operatorname{ker}(f^\infty)=X\), in which case \(f\) is nilpotent, or \(\operatorname{im}(f^\infty)=X\) and \(\operatorname{ker}(f^\infty)=0\), in which case \(f\) is invertible. This proves (ii).

Recall from [Li1] that if \(S\) is a ring and \(S\smallsetminus S^\times\) is a two-sided ideal, then \(S\) is called local.

Corollary. Let \(X\) be a nonzero object in an abelian category satisfying the bi-chain condition. Then \(X\) is indecomposable if and only if \(\operatorname{End}(X)\) is a local ring.

Proof. Lemma reference shows that if \(X\) is indecomposable, every element of \(\operatorname{End}(X)\) is either nilpotent or invertible. The remainder of the argument concerns only the ring structure of \(\operatorname{End}(X)\) and is the same as [Li1].

We can now state the version of the Krull–Remak–Schmidt theorem for abelian categories.

Theorem (M. Atiyah).

Let \(X\) be a nonzero object of an abelian category.

1. If \(X\) satisfies the bi-chain condition, then there exist \(n\in\ensuremath{\mathbb{Z}}_{\geq1}\) and indecomposable subobjects \(X_1,\ldots,X_n\) such that \(X=\bigoplus_{i=1}^nX_i\), and every \(\operatorname{End}(X_i)\) is a local ring.

2. Suppose that \(X\) has decompositions \(\bigoplus_{i=1}^nX_i\) and \(\bigoplus_{j=1}^mX'_j\), where every \(X_i\) and \(X'_j\) is indecomposable and every \(\operatorname{End}(X_i)\) and \(\operatorname{End}(X'_j)\) is local. Then \(n=m\), and there is a bijection \(\sigma\) of \(\{1,\ldots,n\}\) such that \(X_i\simeq X'_{\sigma(i)}\) for every \(i\).

Proof. The main point is to prove that an \(X\) satisfying the bi-chain condition has a decomposition as in (i). Everything else involves only algebraic operations in the rings \(\operatorname{End}\) and is the same as [Li1]. We therefore assume the bi-chain condition in what follows.

If \(X=Y\oplus Z\) with \(Y\neq0\), then \(Y\) also satisfies the bi-chain condition. Indeed, given a bi-chain \((Y_n,\alpha_n,\beta_n)_{n=0}^\infty\) with \(Y_0=Y\), take \(X_n:=Y_n\oplus Z\), \(\tilde{\alpha}_n:=\alpha_n\oplus\ensuremath{\mathrm{id}}_Z\), and \(\tilde{\beta}_n:=\beta_n\oplus\ensuremath{\mathrm{id}}_Z\). This gives a bi-chain with \(X_0=X\), while Lemma reference shows that \(\tilde{\alpha}_n\) (or \(\tilde{\beta}_n\)) is an isomorphism if and only if \(\alpha_n\) (or \(\beta_n\)) is.

If \(X\) is already indecomposable, Corollary reference shows that \(\operatorname{End}(X)\) is local, proving (i). If (i) were false for \(X\), there would be nonzero subobjects \(X_1,Y_1\) such that \(X=X_1\oplus Y_1\) and (i) remained false for \(X_1\). Repeating the process on \(X_1\) gives \(X_1=X_2\oplus Y_2\), and iteration produces a bi-chain \((X_n,\alpha_n,\beta_n)_{n=0}^\infty\), where \(\alpha_n:X_n\twoheadrightarrow X_{n+1}\) and \(\beta_n:X_{n+1}\hookrightarrow X_n\) arise from \(X_n=X_{n+1}\oplus Y_{n+1}\). None is an isomorphism, and \(X_0:=X\), contradicting the bi-chain condition. This proves the assertion.

In applications, the key issue is knowing when the bi-chain condition holds. Here is one sufficient condition.

Proposition. Let \(X\) be an object of finite length in an abelian category (Definition reference). Then \(X\) satisfies the bi-chain condition.

The exercises introduce further ways to deduce the bi-chain condition.

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1. Editorial correction (O014-C024): the source writes \(r:X'\to X\) here, but (ii), the equality \(rf=\ensuremath{\mathrm{id}}_{X'}\), and the subsequent construction all require \(r:X\to X'\). The type-correct direction is used here.↩︎