Formal colimits and compact presentations

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, September 2026. Self-checked; no independent review. Public domain (CC0).

Taking a colimit can erase information about how an object was assembled. A formal colimit retains that information through its maps from test objects. This distinction explains why an infinite set, viewed as one constant object, differs from the formal union of its finite subsets even though both constructions realize the same set.

We assume filtered categories, the universal property of a colimit and Yoneda. Use The Stacks Project's ind-object convention for the basic category and its morphisms. Schapira's Homological Algebra, Section 2.7, gives another account of the formal versus ordinary distinction. Our goal is to understand realization, the two embeddings around it, and the role of objects of finite presentation.

1. Retain the diagram before realizing it

Fix a universe \(\mathcal U\) and a larger ambient universe. The category \(\mathsf C\) is locally \(\mathcal U\)-small; every indexing category below is \(\mathcal U\)-small. The category of its objects may require the larger universe. All colimits and finite-presentation conditions refer to these indexing categories.

An ind-object is a formal filtered diagram, written \(\langle X_i\rangle_{i\in I}\). Its presheaf on \(\mathsf C\) is

\[ T\longmapsto \mathop{\rm colim}_{i\in I} \operatorname{Hom}_{\mathsf C}(T,X_i). \tag{1.1} \]

Isomorphic presheaves determine the same ind-object up to isomorphism. The morphism convention is

\[ \operatorname{Hom}_{\operatorname{Ind}(\mathsf C)} (\langle X_i\rangle,\langle Y_j\rangle) =\mathop{\rm lim}_{i\in I}\mathop{\rm colim}_{j\in J} \operatorname{Hom}_{\mathsf C}(X_i,Y_j). \tag{1.2} \]

These are the definitions in the reference above, rather than an assertion that the ordinary colimit of the diagram exists. Let \(\iota X\) denote the constant diagram at \(X\). Formula (1.2) makes \(\iota\) fully faithful.

Suppose now that \(\mathsf C\) admits all the filtered colimits in question. Realization sends a formal diagram to its actual colimit:

\[ \sigma\langle X_i\rangle=\mathop{\rm colim}_{i}X_i. \]

Proposition 1.1. Realization defines a functor \(\sigma:\operatorname{Ind}(\mathsf C)\to\mathsf C\), with a natural adjunction

\[ \sigma\dashv\iota,\qquad \sigma\iota\simeq\operatorname{id}_{\mathsf C}. \tag{1.3} \]

Proof. A morphism in (1.2) supplies, for each \(i\), a map \(X_i\to Y_j\) for some \(j\), determined up to passage to later stages. Composing with \(Y_j\to\operatorname{colim}_jY_j\) removes the choice of stage. The inverse-limit condition makes these maps compatible with the \(X\)-diagram. The colimit universal property then supplies a map between the realizations. Identities and composites give the expected maps, because they do so on every \(X_i\).

More intrinsically, for each \(Y\) one has natural bijections

\[ \begin{aligned} \operatorname{Hom}_{\operatorname{Ind}(\mathsf C)} (\langle X_i\rangle,\iota Y) &=\mathop{\rm lim}_i\operatorname{Hom}_{\mathsf C}(X_i,Y)\\ &=\operatorname{Hom}_{\mathsf C}(\operatorname{colim}_iX_i,Y). \end{aligned} \]

Thus the realization is determined by the ind-object, independently of its chosen diagram. These bijections give the adjunction. A constant diagram realizes to its value, which proves the second assertion. \(\square\)

The adjunction gives a comparison

\[ \langle X_i\rangle\longrightarrow \iota(\operatorname{colim}_iX_i). \tag{1.4} \]

On a test object \(T\), it takes a map into some stage to the resulting map into the colimit. Surjectivity would say that every map from \(T\) factors through a stage. Injectivity would say that two such factorizations become equal at a common later stage. Neither follows merely from the existence of the colimit.

2. Small pieces that detect maps into a colimit

An object \(D\) is of finite presentation, also called compact with respect to filtered colimits, if for every filtered diagram the canonical map

\[ \mathop{\rm colim}_j\operatorname{Hom}_{\mathsf C}(D,Y_j) \longrightarrow\operatorname{Hom}_{\mathsf C} (D,\operatorname{colim}_jY_j) \tag{2.1} \]

is bijective. This use of “compact” concerns a categorical preservation property, not a topology on \(D\).

Equivalently, for every \(A\in\operatorname{Ind}(\mathsf C)\), the realization comparison

\[ \begin{aligned} \operatorname{Hom}_{\operatorname{Ind}(\mathsf C)}(\iota D,A) &{}\\ \longrightarrow\operatorname{Hom}_{\mathsf C}(D,\sigma A)&{} \end{aligned} \tag{2.F} \]

is bijective. Indeed, if \(A=\langle Y_j\rangle\), its left side is \(\operatorname{colim}_j\operatorname{Hom}_{\mathsf C}(D,Y_j)\) and \(\sigma A=\operatorname{colim}_jY_j\). The comparison is exactly (2.1). Every small filtered diagram defines such an \(A\), and every ind-object has such a presentation, which proves both implications. This equivalence uses realization alone; it requires no generation hypothesis on the finitely presented objects.

Let \(F:\mathsf D\to\mathsf C\) be fully faithful, and suppose every \(F(D)\) is of finite presentation. Neither \(\mathsf D\) nor the full subcategory of all such objects is required to have a small skeleton. Only the individual diagrams used to form ind-objects must be small in the chosen universe.

Theorem 2.1. The realization functor

\[ J_F:\operatorname{Ind}(\mathsf D)\longrightarrow\mathsf C, \qquad \langle D_i\rangle\longmapsto\operatorname{colim}_iF(D_i), \]

is fully faithful. If every object of \(\mathsf C\) is a small filtered colimit of objects in the image of \(F\), then \(J_F\) is an equivalence.

Proof. For diagrams \((D_i)\) and \((E_j)\), compute maps between their realizations by first mapping out of the colimit and then using finite presentation:

\[ \begin{aligned} \operatorname{Hom}_{\mathsf C} (\operatorname{colim}_iF(D_i),\operatorname{colim}_jF(E_j)) &=\mathop{\rm lim}_i \operatorname{Hom}_{\mathsf C}(F(D_i),\operatorname{colim}_jF(E_j))\\ &=\mathop{\rm lim}_i\mathop{\rm colim}_j \operatorname{Hom}_{\mathsf C}(F(D_i),F(E_j))\\ &=\mathop{\rm lim}_i\mathop{\rm colim}_j \operatorname{Hom}_{\mathsf D}(D_i,E_j). \end{aligned} \tag{2.2} \]

This is (1.2), and the bijection is exactly the map induced by realization. It respects composition because each step is natural. Hence \(J_F\) is fully faithful. The additional hypothesis says that every object is isomorphic to a value of \(J_F\), giving essential surjectivity. \(\square\)

Finite presentation controls maps from each piece. The filtered-colimit presentation of all objects controls which objects can be reached. These are distinct hypotheses.

3. Two embeddings with realization between them

Assume that \(\mathsf D\) is a full subcategory of finitely presented objects of \(\mathsf C\), and that every object of \(\mathsf C\) is a small filtered colimit of objects of \(\mathsf D\). By Theorem 2.1, \(J:\operatorname{Ind}(\mathsf D)\to\mathsf C\) is an equivalence. The inclusion of \(\mathsf D\) into \(\mathsf C\) also gives a fully faithful inclusion of their ind-categories, directly by (1.2).

Choose a quasi-inverse to \(J\) and compose it with this inclusion. The resulting functor \(\kappa:\mathsf C\to\operatorname{Ind}(\mathsf C)\) sends \(X\) to a formal presentation of it by the small pieces:

\[ X\simeq\operatorname{colim}_iD_i \quad\Longrightarrow\quad \kappa X\simeq\langle D_i\rangle. \tag{3.1} \]

Different presentations give canonically compatible objects through the equivalence; a presentation is not extra structure imposed on \(X\).

Theorem 3.1. There is an adjoint triple

\[ \kappa\dashv\sigma\dashv\iota. \tag{3.2} \]

Both \(\kappa\) and \(\iota\) are fully faithful, and \(\sigma\kappa\simeq\operatorname{id}\simeq\sigma\iota\). There is a natural comparison \(r_X:\kappa X\to\iota X\). It is an isomorphism exactly when \(X\) is of finite presentation in \(\mathsf C\).

Proof. The construction already gives full faithfulness of \(\kappa\) and \(\sigma\kappa\simeq\operatorname{id}\). For \(X=\operatorname{colim}_iD_i\) and \(A=\langle Y_j\rangle\), finite presentation gives

\[ \begin{aligned} \operatorname{Hom}_{\operatorname{Ind}(\mathsf C)}(\kappa X,A) &=\mathop{\rm lim}_i\mathop{\rm colim}_j \operatorname{Hom}_{\mathsf C}(D_i,Y_j)\\ &=\mathop{\rm lim}_i\operatorname{Hom}_{\mathsf C}(D_i,\sigma A)\\ &=\operatorname{Hom}_{\mathsf C}(X,\sigma A). \end{aligned} \tag{3.3} \]

This proves the first adjunction, naturally in both variables. Proposition 1.1 supplies the other.

The comparison is the unit \(\kappa X\to\iota\sigma\kappa X\) of \(\sigma\dashv\iota\), followed by the identification \(\sigma\kappa X\simeq X\). For every \(A=\langle Y_j\rangle\), precomposition by \(r_X\) is the map

\[ \operatorname{Hom}(\iota X,A)\longrightarrow \operatorname{Hom}(\kappa X,A), \]

which, under (1.2) and (3.3), is precisely (2.1) with \(D=X\). If \(X\) is finitely presented, these maps are all bijective, so Yoneda makes \(r_X\) invertible. If \(r_X\) is invertible, the same computation gives (2.1) for every filtered diagram, proving finite presentation. \(\square\)

Realization therefore has two fully faithful sections with different meanings. The constant section remembers \(X\) as one allowed stage. The other section records its construction from finitely presented pieces.

For sets, take \(\mathsf D\) to be finite sets. A map from a finite set into a filtered colimit uses finitely many elements and finitely many equalities. Filteredness places all those elements and equalities in one stage, proving (2.1). Every set is the filtered union of its finite subsets. Conversely, the identity of an infinite set cannot factor through one of its finite subsets, so an infinite set is not finitely presented. Thus \(\operatorname{Ind}(\mathsf{FinSet})\simeq\mathsf{Set}\).

For \(X=\mathbb N\), the presheaf of \(\kappa X\), tested on a set \(T\), consists of maps \(T\to\mathbb N\) with finite image. The presheaf of \(\iota X\) consists of all such maps. The map \(r_X\) includes the former into the latter. At \(T=\mathbb N\), the identity is missing from the former. Consequently \(r_X\) is not an isomorphism, although \(\sigma r_X\) is an isomorphism of sets.

4. Module presentations over an arbitrary ring

For commutative rings, the algebraic finite-presentation criterion and filtered presentations are already available in Characterizing finite and finitely presented modules. They supply the commutative instance of Theorem 2.1. We give the arbitrary-ring argument to retain the noncommutative instance too.

Let \(R\) be a unital ring, with no commutativity assumption. All modules in this section are left \(R\)-modules. A module is algebraically finitely presented if it has a presentation

\[ R^a\longrightarrow R^b\longrightarrow P\longrightarrow0 \quad(a,b<\infty). \tag{4.1} \]

Theorem 4.1. In \(\mathsf{Mod}(R)\), the objects of finite presentation in the sense of (2.1) are exactly the modules with a presentation (4.1). Every module is a small filtered colimit of such modules. In particular,

\[ \operatorname{Ind}(\mathsf{Mod}_{\rm fp}(R)) \simeq\mathsf{Mod}(R). \tag{4.2} \]

Proof. A homomorphism from a module presented as in (4.1) is determined by the images of \(b\) generators, subject to \(a\) relations. In a filtered colimit, finitely many candidate images can be represented at one stage. Each relation that vanishes in the colimit vanishes at some later stage, and filteredness gives one stage for all the relations. This proves surjectivity in (2.1). If two maps become equal, their values on the finitely many generators become equal at a common later stage. This proves injectivity. The element description used here follows from the usual quotient construction of a module colimit: a finite sum can be moved to a common stage, and an equality involves only finitely many of its defining relations. Left multiplication respects these relations; multiplication of coefficients never needs to be commuted.

To build presentations of an arbitrary module \(M\), use a skeleton of the algebraically finitely presented modules and form the category of all homomorphisms \(P\to M\) from it. It is small: finite matrices over the small ring form a set, and each set of homomorphisms into \(M\) is small. It is nonempty, as \(0\to M\) is an object. Two objects map to their direct sum with the induced map to \(M\).

If \(f,g\colon P\rightrightarrows Q\) are parallel maps over \(M\), their difference has image generated by finitely many elements, since \(P\) is finitely generated. The quotient

\[ Q/\operatorname{im}(f-g) \]

is finitely presented: add those finitely many generators of the image as relations to a finite presentation of \(Q\). Its map to \(M\) is defined because \(f\) and \(g\) agree over \(M\). This quotient equalizes them. Hence the category is filtered.

The canonical map \(\operatorname{colim}_{P\to M}P\to M\) is surjective: every \(m\in M\) is the image of \(1\) under a map \(R\to M\). If an element \(p\in P\) maps to zero in \(M\), the arrow \(P\to P/Rp\), still over \(M\), kills it. The quotient is finitely presented by adding one relation. Any two representatives can first be moved to a common object by filteredness. This proves injectivity and hence the claimed presentation of \(M\).

Finally, suppose \(M\) satisfies (2.1). Apply it to this filtered presentation. The identity of \(M\) factors as \(M\to P\to M\) for some algebraically finitely presented \(P\). Thus \(M\) is a direct summand of \(P\).

We check algebraically that a summand is finitely presented. Write \(P=M\oplus N\). Both summands are finitely generated by projecting a finite generating set of \(P\). Choose finite free surjections \(F_M\to M\) and \(F_N\to N\), giving \(q:F_M\oplus F_N\to P\). Also choose a finite free surjection \(v:G\to P\) whose kernel \(H\) is finitely generated, using (4.1). Freeness supplies lifts \(h:G\to F_M\oplus F_N\) and \(k:F_M\oplus F_N\to G\) with \(qh=v\) and \(vk=q\). Then

\[ \ker q=h(H)+\operatorname{im}(1-hk). \tag{4.3} \]

Indeed, both summands on the right lie in the kernel; for \(z\in\ker q\), the identity \(z=h(kz)+(1-hk)z\) expresses it in that sum, since \(kz\in H\). The right side is finitely generated. But \(\ker q\) is the direct sum of the kernels of \(F_M\to M\) and \(F_N\to N\). Projecting its generators shows that the first of these kernels is finitely generated. This gives a finite presentation of \(M\). The categorical criterion follows, and Theorem 2.1 proves (4.2). \(\square\)

The theorem includes a module \(R/Ru\) over a ring in which \(u\) need not be central. It does not require submodules of finite free modules to be finitely generated. The presentation category handles only finitely many relations at a time.

5. Exercises with solutions

Exercise 1 (introductory: a finite test). Let \(X_n=\{0,\ldots,n\}\) with the inclusion maps, and let \(E\) have three elements. Describe the map

\[ \operatorname{colim}_n\operatorname{Hom}(E,X_n) \longrightarrow\operatorname{Hom}(E,\mathbb N) \]

and prove it is bijective without appealing to Theorem 2.1.

Solution. A map on the left is a function whose three values all lie in one \(X_n\). A function into \(\mathbb N\) has three values; their maximum gives such an \(n\), proving surjectivity. Two stage maps with the same resulting function agree after both are included in \(X_{\max(n,m)}\), proving injectivity. The resulting bijection identifies both sides with ordered triples of natural numbers.

Exercise 2 (intermediate: a noncentral relation). Take \(R=k\langle u,v\rangle\), the free associative algebra on two generators, and \(P=R/Ru\). For a left module \(M\), identify \(\operatorname{Hom}_R(P,M)\). Use your answer to verify finite presentation categorically, keeping the order of multiplication explicit.

Solution. The image \(m\) of the class of \(1\) determines the map \(r+Ru\mapsto rm\). It is well-defined exactly when \(um=0\), since every element of \(Ru\) is a sum of terms \(ru\). Thus the Hom set identifies with \(\{m\in M:um=0\}\). In a filtered colimit, choose a stage representing \(m\). If \(um=0\) in the colimit, that equality holds at some later stage; this gives surjectivity of (2.1). Equality of two choices also holds at a later stage, giving injectivity. No condition involving \(mu\) is meaningful for a general left module. This distinguishes the left ideal \(Ru\) from the right ideal \(uR\).

Exercise 3 (intermediate: finite tests can miss a difference). Let \(r\colon\kappa\mathbb N\to\iota\mathbb N\) be the comparison of Section 3. Show that \(\operatorname{Hom}(\iota E,r)\) is bijective for every finite set \(E\), while \(r\) itself is not an isomorphism. Explain why this does not contradict Theorem 2.1.

Solution. Both Hom sets consist of functions \(E\to\mathbb N\): every such function has finite image. The map between them is the identity on this description. With \(E=\mathbb N\), the identity function lies in \(\operatorname{Hom}(\iota E,\iota\mathbb N)\), but not in \(\operatorname{Hom}(\iota E,\kappa\mathbb N)\). Thus an infinite test detects the failure of invertibility. Theorem 2.1 compares two objects of \(\operatorname{Ind}(\mathsf{FinSet})\) with their realizations. Here \(\iota\mathbb N\) is a constant object of the larger \(\operatorname{Ind}(\mathsf{Set})\); it is not in the image of that smaller ind-category, as the failed comparison shows.

Exercise 4 (advanced: retracts of small pieces). In any category with filtered colimits, let \(M\) be a retract of a finitely presented object \(P\): there are \(a:M\to P\) and \(b:P\to M\) with \(ba=1_M\). Prove that \(M\) is finitely presented, using only the definition (2.1).

Solution. For \(f:M\to\operatorname{colim}_iY_i\), finite presentation of \(P\) factors \(fb\) through a map \(g:P\to Y_i\). Then \(ga:M\to Y_i\) factors \(f\), since \(fba=f\). This proves surjectivity. For injectivity, first put two maps \(u,v:M\to Y_i\) into one common stage. If their colimit maps agree, the maps \(ub,vb:P\to Y_i\) also agree in the colimit. Finite presentation of \(P\) makes them equal after an arrow to a later stage. Precompose with \(a\) and use \(ba=1_M\) to make the images of \(u,v\) equal there. This is precisely equality in the filtered colimit of Hom sets. Both parts of (2.1) hold.

References