nLab additive category

Definition

Definition

An additive category is a category which is

1. an Ab-enriched category;

   (sometimes called a pre-additive category–this means that each hom-set carries the structure of an abelian group and composition is bilinear)

2. which admits finite coproducts

   (and hence, by prop. below, finite products which coincide with the coproducts, hence finite biproducts).

The natural morphisms between additive categories are additive functors.

Remark

A pre-abelian category is an additive category which also has kernels and cokernels. Equivalently, it is an Ab-enriched category with all finite limits and finite colimits. An especially important sort of additive category is an abelian category, which is a pre-abelian one satisfying the extra exactness property that all monomorphisms are kernels and all epimorphisms are cokernels. See at additive and abelian categories for more.

Remark

The Ab-enrichment of an additive category does not have to be given a priori. Every semiadditive category (a category with finite biproducts) is automatically enriched over commutative monoids (as described at biproduct), so an additive category may be defined as a category with finite biproducts whose hom-monoids happen to be groups. (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.) Note that the entire $Ab$-enriched structure follows automatically for abelian categories.

Remark

Some authors use additive category to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a $CMon$-enriched (commutative monoid enriched) category, with or without assumptions of products.

Properties

Proposition

In an Ab-enriched category (or even just a $CMon$-enriched category), a finite product is also a coproduct, and dually (hence a biproduct).

This statement includes the zero-ary case: any terminal object is also an initial object, hence a zero object (and dually), hence every additive category has a zero object.

More precisely, for $\{X_i\}_{i \in I}$ a finite set of objects in an Ab-enriched category, the unique morphism

$$\underset{i \in I}{\coprod} X_i \longrightarrow \underset{j \in I}{\prod} X_j$$

whose components are identities for $i = j$ and are zero otherwise is an isomorphism.

Proof

Consider first the nullary (i.e., zero-ary) case. Given a terminal object $\ast$, the unique morphism $id_\ast: \ast \to \ast$ is the zero morphism $0$ in its hom-object. For any object $A$, the zero morphism $0_A: \ast \to A$ must equal any morphism $f: \ast \to A$ on account of $f = f id_\ast = f 0 = 0_A$ where the last equation is by $CMon$-enrichment. Hence $\ast$ is initial. (N.B.: this argument applies more generally to categories enriched in pointed sets, and is self-dual.)

Consider now the case of binary (co-)products. Using zero morphisms, in addition to its canonical projection maps $p_i \colon X_1 \times X_2 \to X_i$, any binary product also admits “injection” maps $X_i \to X_1 \times X_2$, and dually for the coproduct:

$$\array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{p_{X_1}}} && {}_{\mathllap{p_{X_2}}}\searrow \\ X_1 && && X_2 } \;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\; \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{i_{X_1}}} && {}^{\mathllap{i_{X_2}}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \sqcup X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow \\ X_1 && && X_2 } \,.$$

Observe some basic compatibility of the $Ab$-enrichment with the product:

First, for $(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2$ then

$$(\star) \;\;\;\;\;\; (\alpha_1,\beta_1) + (\alpha_2, \beta_2) = (\alpha_1+ \alpha_2 , \; \beta_1 + \beta_2)$$

(using that the projections $p_1$ and $p_2$ are linear and by the universal property of the product).

Second, $(id,0) \circ p_1$ and $(0,id) \circ p_2$ are two projections on $X_1\times X_2$ whose sum is the identity:

$$(\star\star) \;\;\;\;\;\; (id, 0) \circ p_1 + (0, id) \circ p_2 = id_{X_1 \times X_2} \,.$$

(We may check this, via the Yoneda lemma on generalized elements: for $(\alpha, \beta) \colon R \to X_1\times X_2$ any morphism, then $(id,0)\circ p_1 \circ (\alpha,\beta) = (\alpha,0)$ and $(0,id)\circ p_2\circ (\alpha,\beta) = (0,\beta)$, so the statement follows with equation $(\star)$.)

Now observe that for $f_i \;\colon\; X_i \to Q$ any two morphisms, the sum

$$\phi \;\coloneqq\; f_1 \circ p_1 + f_2 \circ p_2 \;\colon\; X_1 \times X_2 \longrightarrow Q$$

gives a morphism of cocones

$$\array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & && \\ X_1 && \downarrow^{\mathrlap{\phi}} && X_2 \\ & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}} \\ && Q } \,.$$

Moreover, this is unique: suppose $\phi'$ is another morphism filling this diagram, then, by using equation $(\star \star)$, we get

$$\begin{aligned} \phi & = \phi \circ id_{X_1 \times X_2} \\ &= \phi \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 ) \\ & = \phi \circ (id_{X_1}, 0) \circ p_1 + \phi \circ (0, id_{X_2}) \circ p_2 \\ & = f_1 \circ p_1 + f_2 \circ p_2 \\ & = \phi' \circ (id_{X_1}, 0) \circ p_1 + \phi' \circ (0, id_{X_2}) \circ p_2 \\ & = \phi' \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 ) \\ &= \phi' \circ id_{X_1 \times X_2} \\ &= \phi' \end{aligned} \,.$$

This means that $X_1\times X_2$ satisfies the universal property of a coproduct.

By a dual argument, the binary coproduct $X_1 \sqcup X_2$ is seen to also satisfy the universal property of the binary product. By induction, this implies the statement for all finite (co-)products. (If a particular finite (co-)product exists but binary ones do not, one can adapt the above argument directly to that case.)

Remark

Such products which are also coproducts as in prop. are sometimes called biproducts or direct sums; they are absolute limits for Ab-enrichment.

Remark

The coincidence of products with biproducts in prop. does not extend to infinite products and coproducts.) In fact, an Ab-enriched category is Cauchy complete just when it is additive and moreover its idempotents split.

Conversely:

Definition

A semiadditive category is a category that has all finite products which, moreover, are biproducts in that they coincide with finite coproducts as in def. .

Proposition

In a semiadditive category, def. , the hom-sets acquire the structure of commutative monoids by defining the sum of two morphisms $f,g \;\colon\; X \longrightarrow Y$ to be

$$f + g \;\coloneqq\; X \overset{\Delta_X}{\to} X \times X \simeq X \oplus X \overset{f \oplus g}{\longrightarrow} Y \oplus Y \simeq Y \sqcup Y \overset{\nabla_Y}{\to} Y \,.$$

With respect to this operation, composition is bilinear.

Proof

The associativity and commutativity of $+$ follows directly from the corresponding properties of $\oplus$. Bilinearity of composition follows from naturality of the diagonal $\Delta_X$ and codiagonal $\nabla_X$:

$$\array{ W &\overset{\Delta_W}{\longrightarrow}& W \times W &\overset{\simeq}{\longrightarrow}& W \oplus W \\ \downarrow^{\mathrlap{e}} && \downarrow^{\mathrlap{e \times e}} && \downarrow^{\mathrlap{e \oplus e}} \\ X &\overset{\Delta_X}{\to}& X \times X &\simeq& X \oplus X &\overset{f \oplus g}{\longrightarrow}& Y \oplus Y &\simeq& Y \sqcup Y &\overset{\nabla_X}{\to}& Y \\ && && && \downarrow^{\mathrlap{h \oplus h}} && \downarrow^{\mathrlap{h \sqcup h}} && \downarrow^{\mathrlap{h}} \\ && && && Z \oplus Z &\simeq& Z \sqcup Z &\overset{\nabla_Z}{\to}& Z }$$

Proposition

Given an additive category according to def. , then the enrichment in commutative monoids which is induced on it via prop. and prop. from its underlying semiadditive category structure coincides with the original enrichment.

Proof

By the proof of prop. , the codiagonal on any object in an additive category is the sum of the two projections:

$$\nabla_X \;\colon\; X \oplus X \overset{p_1 + p_2}{\longrightarrow} X \,.$$

Therefore (checking on generalized elements, as in the proof of prop. ) for all morphisms $f,g \colon X \to Y$ we have commuting squares of the form

$$\array{ X &\overset{f+g}{\longrightarrow}& Y \\ {}^{\mathllap{\Delta_X}}\downarrow && \uparrow^{\mathrlap{\nabla_Y =}}_{\mathrlap{p_1 + p_2}} \\ X \oplus X &\underset{f \oplus g}{\longrightarrow}& Y\oplus Y } \,.$$

Remark

Prop. says that being an additive category is an extra property on a category, not extra structure. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.

Related concepts

• semiadditive category

• abelian category

• additive (∞,1)-category

• triangulated category

References

Textbook accounts:

• Masaki Kashiwara, Pierre Schapira, Section 8 of: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

Discussion in homological algebra:

• Charles Weibel, An Introduction to Homological Algebra, Cambridge University Press (1994) [doi:10.1017/CBO9781139644136, pdf]

See also:

• William Lawvere, Introduction to Linear Categories and Applications, course lecture notes (1992) [pdf, pdf]

Discussion of model category structures on additive categories is around def. 4.3 of

• Apostolos Beligiannis, Homotopy theory of modules and Gorenstein rings, Math. Scand. 89 (2001) (pdf)

Formalization of additive categories as univalent categories in univalent foundations of mathematics (homotopy type theory):

• unimath, Additive categories [UniMath.CategoryTheory.Additive]

A characterisation of preadditive categories in terms of commutative monoids in cartesian multicategories is given in:

• Claudio Pisani, Sequential multicategories, Theory and Applications of Categories 29.19 (2014), arXiv:1402.0253