### Context

#### Enriched category theory

Could not include enriched category theory - contents

## Derived categories

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

###### Definition

A functor $F:𝒜\to ℬ$ between additive categories is itself called additive if it preserves finite biproducts.

That is,

1. $F$ maps a zero object to a zero object, $F\left(0\right)\simeq 0\in ℬ$;

2. given any two objects $x,y\in 𝒜$, there is an isomorphism $F\left(x\oplus y\right)\cong F\left(x\right)\oplus F\left(y\right)$, and this respects the inclusion and projection maps of the direct sum:

$\begin{array}{ccccc}x& & & & y\\ & {}_{{i}_{X}}↘& & {↙}_{{i}_{y}}\\ & & x\oplus y\\ & {}^{{p}_{x}}↙& & {↘}^{{p}_{y}}\\ x& & & & y\end{array}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\stackrel{F}{↦}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\begin{array}{ccccc}F\left(x\right)& & & & F\left(y\right)\\ & {}_{{i}_{F\left(x\right)}}↘& & {↙}_{{i}_{F\left(y\right)}}\\ & & F\left(x\oplus y\right)\cong F\left(x\right)\oplus F\left(y\right)\\ & {}^{{p}_{F\left(X\right)}}↙& & {↘}^{{p}_{F\left(y\right)}}\\ F\left(x\right)& & & & F\left(y\right)\end{array}$\array { x & & & & y \\ & {}_{\mathllap{i_X}}\searrow & & \swarrow_{\mathrlap{i_y}} \\ & & x \oplus y \\ & {}^{\mathllap{p_x}}\swarrow & & \searrow^{\mathrlap{p_y}} \\ x & & & & y } \quad\quad\stackrel{F}{\mapsto}\quad\quad \array { F(x) & & & & F(y) \\ & {}_{\mathllap{i_{F(x)}}}\searrow & & \swarrow_{\mathrlap{i_{F(y)}}} \\ & & F(x \oplus y) \cong F(x) \oplus F(y) \\ & {}^{\mathllap{p_{F(X)}}}\swarrow & & \searrow^{\mathrlap{p_{F(y)}}} \\ F(x) & & & & F(y) }
###### Remark

In practice, functors between additive categories are generally assumed to be additive.

## Examples

###### Example

The hom-functor $\mathrm{Hom}\left(-,-\right):{𝒜}^{\mathrm{op}}×𝒜\to \mathrm{Ab}$ is additive (and in both arguments separately).

###### Example

For $𝒜=R$Mod and $N\in 𝒜$, the functor that forms tensor product of modules $\left(-\right)\otimes N:𝒜\to 𝒜$.

In fact thes examples are generic, see prop. 2 below.

## Properties

### Relation to $\mathrm{Ab}$-enriched functors

An additive category canonically carries the structure of an Ab-enriched category where the $\mathrm{Ab}$-enrichment structure is induced from the biproducts as described at biproduct.

###### Proposition

With respect to the canonical Ab-enriched category-structre on additive categories $𝒜$, $ℬ$, additive functors $F:𝒜\to ℬ$ are equivalently Ab-enriched functors.

###### Proof

An $\mathrm{Ab}$-enriched functor preserves all finite biproducts that exist, since finite biproducts in Ab-enriched categories are Cauchy colimits.

### Characterization of right exact additive functors

Let $R,R\prime$ be rings.

The following is the Eilenberg-Watts theorem. See there for more.

###### Proposition

If an additive functor $F:R$Mod $\to R\prime$Mod is a right exact functor, then there exists an $R\prime$-$R$-bimodule $B$ and a natural isomorphism

$F\simeq B{\otimes }_{R}\left(-\right)$F \simeq B \otimes_R (-)

with the functor that forms the tensor product with $B$.

This is (Watts, theorem 1),

## References

• Charles Watts?, Intrinsic characterizations of some additive functors, Proceedings of the American Mathematical Society (1959) (JSTOR)

Revised on April 3, 2013 01:12:01 by Urs Schreiber (82.169.65.155)