Formal linear combinations and finite sums

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, October 2026. Self-checked by GPT-6.1 Sol, the AI that wrote it; no independent review. Public domain (CC0).

A category may specify objects and maps without specifying sums of maps or sums of objects. We can add these operations in two steps. First take formal integer combinations of maps. Then take finite lists of objects and matrices between them. A functor into an additive category extends across both steps, and its natural transformations extend with it.

The two operations do different jobs. Formal combinations make each group of maps additive. Finite lists supply the objects needed for biproducts. Neither operation promises to split every idempotent. We will see this distinction in a concrete example.

We assume the definitions of categories, functors, natural transformations and abelian groups. For projector splittings, read Retracts and stabilization of formal objects, Section 1. Retain the complete binary additive-functor criterion in the AI-integrated Stacks baseline. Its zero-map and biproduct interfaces are fully proved in Recovering addition from products and coproducts, Section 1. Other references are Mathlib's linear completion, matrix categories, and additive adjunctions.

A category is preadditive when its groups of maps are abelian and composition is bilinear. It is additive when it also has a zero object and finite biproducts. An additive functor induces homomorphisms on groups of maps. Categories are locally small; object collections may belong to a larger universe. All lists in this lesson are finite. The presheaf description in Section 5 additionally assumes that the original category is small.

1. Recognizing addition from products

For a biproduct \(X_1\oplus X_2\), denote its inclusions by \(i_1,i_2\) and projections by \(p_1,p_2\). Its identities are

These equations also characterize a biproduct in a preadditive category: retain both explicit inverse maps and uniqueness proofs in Recovering addition, Section 1. That section retains the complete canonical zero-object equivalence and proves that the ordinary zero maps factoring through zero are precisely the enriched zero maps. Thus every zero map used below has the required factorization, and every use of (1.1) gives both actual universal structures.

We retain the following stronger binary-product criterion from the complete canonical additive-functor proof, with the full product-to-coproduct proof and the coordinate interfaces just specified.

Theorem 1.1. For a functor \(F\) between additive categories, the following are equivalent: it is additive; its canonical binary-product comparisons are isomorphisms; its canonical binary-coproduct comparisons are isomorphisms. Each condition implies preservation of all finite products and coproducts, including zero.

The empty case requires no separate assumption: the product comparison at \((0,0)\) makes the diagonal of \(F0\) an isomorphism. Every group of maps into \(F0\) then has a bijective diagonal and is trivial, so \(F0\) is zero.

Lemma 1.2. If the first projection \(p_X:X\oplus Y\to X\) is invertible, then \(Y\) is zero.

Proof. Its section \(i_X\) must be its inverse. Thus \(p_Y=p_Yi_Xp_X=0\), and \(1_Y=p_Yi_Y=0\). Apply the zero-object criterion. \(\square\)

Proposition 1.3. Every fully faithful functor between additive categories is additive.

Proof. Full faithfulness gives only one endomorphism of \(F0\), so its identity is zero and \(F0\) is zero. Every zero map factors through zero, so \(F\) preserves zero maps.

For a source biproduct \(S=X_1\oplus X_2\), put \(e=F(i_1)F(p_1)+F(i_2)F(p_2)\). Fullness gives an endomorphism \(h:S\to S\) with \(Fh=e\). From the first equations in (1.1) and preservation of zero,

Faithfulness gives \(p_rh=p_r\). The source product property gives \(h=1_S\), hence \(e=1_{FS}\). All equations (1.1) now hold for the images of the structure maps. They specify an actual biproduct in the target. Theorem 1.1 proves additivity. \(\square\)

Faithfulness alone does not suffice. The functor \(\mathsf{Ab}\to\mathsf{Ab}\) taking a group to the free abelian group on its underlying set is faithful: basis elements distinguish two different group maps. But it sends the zero group to \(\mathbb Z\), so it is not additive.

2. Why an adjoint inherits addition

Adjunctions do not initially assert that their Hom bijections preserve addition. They acquire this property when either adjoint is additive.

We retain the preadditive theorem in Mathlib, Additive adjunctions, which requires no zero object or biproducts.

Theorem 2.1. Let \(F:\mathsf B\to\mathsf A\) be left adjoint to \(G:\mathsf A\to\mathsf B\), with both categories preadditive. Then \(F\) is additive if and only if \(G\) is additive. In that case the adjunction bijections are isomorphisms of abelian groups.

Here is the identity check behind the interface. With unit \(\eta\) and counit \(\epsilon\), the bijection and its inverse are

If \(F\) is additive, the inverse is additive. For \(a,b:Y\to Z\), use the bijection \(\Phi_{GY,Z}\). Counit naturality gives

Injectivity gives additivity of \(G\). The opposite adjunction proves the converse. Formula (2.1) then makes \(\Phi\) additive too. Additivity includes preservation of zero and negatives since the groups are abelian.

This theorem concerns addition of maps. Preservation of kernels or cokernels requires further hypotheses; Exercise 4 separates these properties.

For the coefficient category in Exercise 4, retain the complete module kernel and cokernel constructions in Kernels, cokernels and the abelian comparison, Section 5, together with the compatible coimage-to-image bijection and its linear inverse in Coimages, Section 5. Specialize to the field \(k\). A kernel is a subspace of its finite-dimensional source and a cokernel is a quotient of its finite-dimensional target; both remain finite-dimensional. Zero and finite direct sums do too. Thus these universal properties and the invertible comparison restrict to the coefficient category in that exercise. For its two-object arrow diagram, retain the full naturality, kernel, cokernel and inverse-comparison proofs in Pointwise abelian structure and natural splittings, Section 2. This supplies the endpoint universal properties and the detection of exactness used in its solution.

3. First add combinations of maps

Let \(\mathsf C\) be any locally small category. Define \(\mathbb Z[\mathsf C]\) to have the same objects, with

The notation on the right means finite-support integer combinations of symbols \([f]\). Distinct maps give distinct symbols; an empty set of maps gives the zero group. Define composition by bilinear extension of

Every composition involves finite sums. On basis symbols, associativity is associativity in \(\mathsf C\); expansion by bilinearity proves it on all combinations. The identity laws follow the same way. Thus this is a preadditive category, and \(j:\mathsf C\to\mathbb Z[\mathsf C]\), \(f\mapsto[f]\), is a faithful ordinary functor.

The generic linear construction is Mathlib's \(\operatorname{Free}(R,\mathsf C)\), defined in the later CategoryTheory.Free section of the cited module, after its free-module construction. Here \(R=\mathbb Z\). To check the target hypothesis, every abelian group has its canonical integer action: nonnegative integers act by repeated addition, and negative integers act by negatives. Induction on nonnegative integers, followed by taking negatives, shows that a group homomorphism preserves this action. Bilinearity therefore makes any preadditive category \(\mathbb Z\)-linear, and makes every additive functor \(\mathbb Z\)-linear. Conversely, a \(\mathbb Z\)-linear functor is additive by its group-homomorphism requirement. Hence Mathlib's linear target and linear lift specialize to exactly the preadditive target and additive lift below. The formulas prove the required universal property, including all natural transformations, directly.

Proposition 3.1. For every functor \(F:\mathsf C\to\mathsf A\) into a preadditive category, there is a unique additive functor \(\widehat F:\mathbb Z[\mathsf C]\to\mathsf A\) with \(\widehat Fj=F\), using the same object values. Its map formula is

Every natural transformation \(F\to G\) extends uniquely to \(\widehat F\to\widehat G\).

Proof. Formula (3.3) is well-defined on the free abelian groups. Bilinearity in the target shows that it respects (3.2) and all composites of finite sums. It preserves identities and addition. Conversely, an additive extension must send each basis symbol to \(F(f)\), so (3.3) is forced.

Keep the components of a natural transformation. Naturality on a combination follows by taking the integer combination of its naturality equations on the individual maps. Uniqueness holds because the objects have not changed. \(\square\)

Only the groups of maps have changed. For example, a category with one object and only its identity becomes a one-object preadditive category whose endomorphism ring is \(\mathbb Z\). It still has no zero object: the sole object's identity is \(1\), not zero.

4. Then add finite sums of objects

Let \(\mathsf D\) be preadditive. Define \(\operatorname{Mat}(\mathsf D)\) to have objects finite lists \((X_1,\ldots,X_n)\), including the empty list. A map from this list to \((Y_1,\ldots,Y_m)\) is a matrix \(a=(a_{ri})\), where \(a_{ri}:X_i\to Y_r\) belongs to \(\mathsf D\). Matrix addition is entrywise, and composition is

The identity is the diagonal matrix of identities, with zero elsewhere. Bilinearity, the identity laws and associativity follow by finite expansion; associativity interchanges two finite summations and uses associativity in \(\mathsf D\). The empty list is zero, and concatenation is a biproduct. Its block inclusions and projections satisfy (1.1). Thus \(\operatorname{Mat}(\mathsf D)\) is additive.

The singleton functor \(s:\mathsf D\to\operatorname{Mat}(\mathsf D)\) is fully faithful and additive. Retain the matrix category, finite biproducts, embedding and additive lift in Mathlib, Matrix categories. Here is the exact index interface. Mathlib uses a finite index set \(I\) and a family \(X:I\to\mathsf D\). A list uses its finite set of positions; its entry \(a_{ri}:X_i\to Y_r\) is Mathlib's entry with source index \(i\) and target index \(r\). Both composition conventions give \(\sum_r b_{sr}a_{ri}\). Conversely choose a bijection from the positions of a list of length \(|I|\) to \(I\), and list the corresponding objects. The reindexing map has one identity entry for each matched index and zeros elsewhere; reversing the bijection gives its inverse. Matrix multiplication leaves one identity in each diagonal position and zero elsewhere, so both composites are identities. Reindexing preserves addition and all finite sums in composition. The list model is therefore fully faithful and essentially surjective onto the finite-index model, including the empty family. It preserves their specified finite biproducts. No infinite-index construction or extra object-size hypothesis is being imported.

Proposition 4.1. An additive functor \(H:\mathsf D\to\mathsf A\), with \(\mathsf A\) additive, extends to an additive functor \(H^\oplus:\operatorname{Mat}(\mathsf D)\to\mathsf A\). It sends a list to the chosen biproduct \(\bigoplus_iH(X_i)\), and a matrix to the map whose entries are \(H(a_{ri})\).

Proof. Choose finite biproducts, including zero, in the target. With target inclusions \(\iota_r\) and source projections \(\pi_i\), the map is

The equations \(\pi_r\iota_t=\delta_{rt}\) collapse the composite of two such sums to the matrix (4.1). Additivity of \(H\) identifies its entries with \(H(\sum_r b_{sr}a_{ri})\). The sum of diagonal blocks is the identity on a biproduct; the empty sum is the identity of zero. This proves the functor laws. Entrywise addition proves additivity. \(\square\)

The singleton values are canonically isomorphic to \(H(X)\). More generally, every additive functor \(L:\operatorname{Mat}(\mathsf D)\to\mathsf A\) preserves the displayed finite biproducts. Hence it is determined, up to a natural isomorphism compatible with its singleton values, by \(Ls\).

We need the stronger assertion about all transformations.

Lemma 4.2. Restriction along \(s\) is fully faithful on additive functors into \(\mathsf A\).

Proof. Suppose \(\alpha:Ls\to Ms\) is a natural transformation. On a list \(X=(X_1,\ldots,X_n)\), define

where \(i_i,p_i\) are its singleton inclusions and projections. On the empty list use the unique map between zero objects.

For a matrix \(a:X\to Y\), the \((r,i)\) entry of the equation \(\alpha_YL(a)=M(a)\alpha_X\) is exactly naturality of the given transformation for \(a_{ri}:X_i\to Y_r\). The images of the structure maps are biproduct maps, so equality of all entries proves that equation. We have extended \(\alpha\).

Any extension must have (4.3): naturality with the inclusions fixes its restrictions to every source summand, and those restrictions determine the map. Thus restriction is full and faithful. Componentwise composition and identities are retained. \(\square\)

Combining Propositions 3.1 and 4.1 with Lemma 4.2 gives an equivalence of functor categories

Here all natural transformations are allowed. The left side requires additive functors, while the right side contains ordinary functors. Existence uses chosen finite biproducts; full faithfulness is the explicit transformation argument above. In particular, once two extensions and their comparisons with a fixed \(F\) are specified, there is exactly one natural isomorphism respecting those comparisons.

These constructions use only finite lists and the original groups of maps. They do not impose global smallness of \(\mathsf C\) or \(\mathsf A\). If their object and Hom universes differ, place them in a common larger universe; the formulas and the original Hom sets do not change.

5. A realization by abelian presheaves

Assume now that \(\mathsf C\) is small. For \(X\in\mathsf C\), define an abelian presheaf

Restriction along \(a:T'\to T\) sends \([f]\) to \([fa]\). Postcomposition gives a functor \(P:\mathsf C\to\operatorname{Fun}(\mathsf C^{\mathrm{op}},\mathsf{Ab})\).

Lemma 5.1. For any abelian presheaf \(A\), evaluation on \([1_X]\) gives a group isomorphism

These natural transformations have group homomorphisms as components.

Proof. For \(a\in A(X)\), set

It is a group homomorphism. Restriction along \(u:T'\to T\) changes the right side to \(\sum_f n_fA(fu)(a)\), which is also \(\theta_{T'}(\sum_f n_f[fu])\); this proves naturality.

Conversely, naturality with \(f:T\to X\) forces \(\theta_T([f])=A(f)(\theta_X([1_X]))\). Linearity forces (5.3) on every element. These two constructions are inverse and additive. \(\square\)

In particular,

Under this isomorphism, composition is bilinear extension of \([g][f]=[gf]\), by (5.3). Hence \(X\mapsto P_X\) realizes \(\mathbb Z[\mathsf C]\) fully faithfully. The original functor \(P\) is faithful: evaluating \(P(f)\) on \([1_X]\) gives \([f]\), and different \(f\) are different basis symbols.

Theorem 5.2. Let \(\mathsf C^+\) be the full subcategory of abelian presheaves formed by finite products of the \(P_X\), together with their isomorphic copies. It is additive, and

Every functor \(F:\mathsf C\to\mathsf A\) into an additive category extends to an additive functor \(F^+:\mathsf C^+\to\mathsf A\), with a specified natural isomorphism \(F^+P\simeq F\). Restriction along \(P\) is an equivalence from additive functors on \(\mathsf C^+\) to ordinary functors on \(\mathsf C\).

Proof. Finite products and sums of abelian groups agree, pointwise. The empty product is the zero presheaf. Concatenating two finite families shows that \(\mathsf C^+\) has finite biproducts; its groups of maps and bilinear composition come from the abelian presheaf category.

Send a list \((X_i)\) to \(\bigoplus_iP_{X_i}\). A map between two such sums is determined by its entries \(P_{X_i}\to P_{Y_r}\). Lemma 5.1 identifies each entry with a finite integer combination of maps \(X_i\to Y_r\). Formula (5.3) verifies that entrywise composition is (4.1). This functor is thus fully faithful. It is essentially surjective by the definition of \(\mathsf C^+\), proving (5.5).

Choose a quasi-inverse for (5.5), then apply (4.4). The resulting \(F^+\) and comparison give the asserted extension. The restriction equivalence, including every natural transformation, follows from the same composition of equivalences. If only the displayed finite products are chosen as objects, omitting other isomorphic copies, the equivalent full subcategory has the same conclusion. \(\square\)

This model makes the two steps visible. Individual \(P_X\) already carry formal integer combinations of maps. Their finite sums carry the matrix operations. It adds no completion under arbitrary colimits or projector images.

6. Four exercises with complete solutions

Exercise 1 (warm-up: the arrow category as triangular matrices). Let \(\mathsf C\) have two objects \(a,b\), their identities, and a single other map \(u:a\to b\). Write \((a^m,b^n)\) for a list with \(m\) copies of \(a\) and \(n\) copies of \(b\). Describe all maps from \((a^m,b^n)\) to \((a^p,b^q)\) in its free additive completion. Give the composition formula. If \(F\) assigns abelian groups \(A,B\) and a homomorphism \(h:A\to B\) to these data, describe its extension on every matrix.

Solution. In \(\mathbb Z[\mathsf C]\), the endomorphism groups of \(a,b\) are \(\mathbb Z\), the group from \(a\) to \(b\) is \(\mathbb Z[u]\), and the group from \(b\) to \(a\) is zero. Here \(\mathbb Z[u]\) means the free abelian group on the single symbol \(u\), not a polynomial ring. Thus a map has the block form

The entries of \(V\) are coefficients of \(u\). Composing with a matrix \(\left(\begin{smallmatrix}U'&0\\V'&W'\end{smallmatrix}\right)\) gives

Only identity compositions and compositions of \(u\) with identities occur, so this is ordinary block multiplication with the indicated integer coefficients.

The extended object is \(A^m\oplus B^n\). Its map has blocks \(U\) acting by integer multiples of \(1_A\), \(W\) by integer multiples of \(1_B\), and \(V\) by integer multiples of \(h\); its upper-right block is zero. Bilinearity gives (6.2) after applying \(F\), as required. Empty rows or columns describe zero objects and zero maps without additional conventions. \(\square\)

Exercise 2 (intermediate: finite sums need not split projectors). Let \(\mathsf C\) have one object and two maps \(1,e\), with \(e^2=e\) and \(e\ne1\). Identify its linearized endomorphism ring with

Show that its free additive completion is equivalent to the category of finite free \(R\)-modules. Prove that multiplication by \(t\) on \(R\) has no splitting inside this category.

Solution. The two basis symbols \(1,e\) freely generate the endomorphism group. Their multiplication is \(1\) as identity and \(e^2=e\), so \(e\mapsto t\) gives the first ring description. The map

is a ring homomorphism to \(\mathbb Z\times\mathbb Z\). It is bijective: \((x,y)\) has preimage \(x+(y-x)t\). This proves (6.3), without any coprimality assumption.

A list of length \(n\) has maps given by matrices over \(R\). Sending it to \(R^n\) identifies matrix composition and all module homomorphisms between these finite free modules, proving the equivalence. Commutativity of \(R\) makes left and right module conventions agree here.

Under (6.4), multiplication by \(t\) is the projector with image \(0\times\mathbb Z\). If it split through a finite free module \(R^n\), the splitting inclusion would identify \(R^n\) with that image: \(ab=t\), \(ba=1\) imply that the image of \(a\) equals the image of \(t\). Multiplying this alleged isomorphism by the central projector \((1,0)\) gives an isomorphism \(\mathbb Z^n\simeq0\). Thus \(n=0\), since a nonempty free group has a nonzero basis vector. But the alleged image has a nonzero second component. This contradiction proves the nonsplitting. The free additive completion is therefore not idempotent complete. \(\square\)

Exercise 3 (hard: all transformations, and what uniqueness means). Take \(\mathsf C\) to have one object and only its identity. For an abelian group \(A\), let \(E_A\) be the additive functor from its free additive completion to \(\mathsf{Ab}\) sending a list of length \(n\) to \(A^n\), and integer matrices to their usual actions. Prove

Determine the natural automorphisms of \(E_{\mathbb Z}\). Explain why a fixed identity comparison on the one-object category singles out exactly one automorphism.

Solution. A group homomorphism \(v:A\to B\) gives components \(v^n:A^n\to B^n\). They commute with every integer matrix since \(v(rx+sy)=rv(x)+sv(y)\). The component at zero is the unique map of zero groups. Thus they form a natural transformation.

Conversely, its component at the singleton is a homomorphism \(v:A\to B\). Naturality for the \(n\) singleton inclusions forces the \(n\)-th component to restrict to \(v\) on each corresponding summand, with zero in the other coordinates. Those restrictions determine the map from the biproduct, so the component is exactly \(v^n\). This proves (6.5).

Every endomorphism of \(\mathbb Z\) is multiplication by its value on \(1\), an integer \(r\). It is invertible exactly when some integer \(s\) has \(rs=1\), hence exactly for \(r=1\) and \(r=-1\). Both give distinct natural automorphisms of \(E_{\mathbb Z}\). Therefore an unrestricted isomorphism between two extensions need not be unique. Requiring compatibility with the identity comparison on the singleton forces \(v=1_{\mathbb Z}\); the preceding determination forces the identity on every list. This is the precise uniqueness used in (4.4) and Theorem 5.2. \(\square\)

Exercise 4 (advanced: an adjunction chain for arrows). Let \(k\) be a field, and let \(\mathsf V\) be the category of finite-dimensional \(k\)-spaces. Its arrow category has objects \(h:X_0\to X_1\) and maps commuting squares. Define

Show that there is a chain of adjunctions

Identify every Hom bijection and verify its naturality. Show that all seven functors are additive, but \(\operatorname{coker}\) is not left exact and \(\ker\) is not right exact.

Solution. A square from \(h:X_0\to X_1\) to \(h':Y_0\to Y_1\) is a pair \((u,v)\) with \(h'u=vh\). This description gives the following six bijections.

The inverse in each row either inserts the forced component or uses the kernel inclusion or cokernel projection. Each is uniquely determined. Composing squares composes the selected component; the induced kernel and cokernel maps are defined by their universal properties. Thus the bijections commute with precomposition and postcomposition, proving their naturality and all adjunctions in (6.7).

The arrow category is additive: add squares componentwise, and take zero and finite biproducts at both endpoints. The functors \(S,D,R,e_0,e_1\) preserve those group additions directly. Theorem 2.1 now gives additivity of \(\operatorname{coker}\) and \(\ker\) as well.

Kernels and cokernels of squares are formed at both endpoints; the equation \(h'u=vh\) induces the required map between the endpoint kernels or cokernels. Their componentwise universal properties give the square universal properties. The ordinary coimage-to-image comparison is an isomorphism at both endpoints, so the arrow category is abelian. In particular the following endpointwise short exact sequence is short exact in it:

The first square is \((0,1_k)\), and the second is \((1_k,0)\). Applying \(\operatorname{coker}\) gives \(0\to k\to0\to0\to0\), which is not exact at \(k\). Applying \(\ker\) gives \(0\to0\to0\to k\to0\), also not exact at its nonzero term. This proves the claimed failures while keeping additivity and all adjunctions intact. \(\square\)

References

• The Stacks Project Authors, AI-integrated fork at the linked immutable revision, Homology: complete zero-object proof, product-to-coproduct proof and binary additive-functor proof. GFDL 1.2 or later; referenced without copying source text. The exact zero-factorization and both coordinate universals are retained from the complete owned addition lesson.
• Mathlib source attribution: the free-module and later free linear category constructions, and the matrix category, have copyright © Kim Morrison 2021; the free-module module also credits Johan Commelin. Additive adjunctions have copyright © Sophie Morel 2024 and credit Sophie Morel and Joël Riou. These Apache-2.0 sources are referenced without copying code. The full arguments and all transformations required here are given above.
• The complete owned module and coimage proofs are retained through the links in Section 2. The linked pointwise lesson remains CC BY-NC-SA 4.0, including its Tom Leinster attribution and disclaimer; no text from that lesson is copied here.
• The Stacks Project. Preadditive and additive categories and Additive functors. We use the zero and biproduct criteria and the binary preservation characterization of additivity.
• The Mathlib Community. Free linear categories, Matrix categories and additive envelopes, and Additive adjunctions.