Contents

### Context

#### Enriched category theory

enriched category theory

## Derived categories

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Definition

###### Definition

An additive category is a category which is

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

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

###### 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.

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') & = (\phi - \phi') \circ id_{X_1 \times X_2} \\ &= (\phi - \phi') \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 ) \\ & = \underset{ = 0}{\underbrace{(\phi - \phi') \circ (id_{X_1}, 0)}} \circ p_1 + \underset{ = 0}{\underbrace{(\phi - \phi') \circ (0, id_{X_2})}} \circ p_2 \\ & = 0 \end{aligned}

and hence $\phi = \phi'$. 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

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

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 enrichement 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.

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)