Could not include enriched category theory - contents
additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
and
nonabelian homological algebra
A functor $F: \mathcal{A} \to \mathcal{B}$ between additive categories is itself called additive if it preserves finite biproducts.
That is,
$F$ maps a zero object to a zero object, $F(0) \simeq 0 \in \mathcal{B}$;
given any two objects $x, y \in \mathcal{A}$, there is an isomorphism $F(x \oplus y) \cong F(x) \oplus F(y)$, and this respects the inclusion and projection maps of the direct sum:
In practice, functors between additive categories are generally assumed to be additive.
Each of the following conditions is sufficient for guaranteeing that a functor $\mathcal{A} \to \mathcal{B}$ preserves biproducts (where $\mathcal{A}$ and $\mathcal{B}$ are categories with a zero object):
The hom-functor $Hom(-,-) \colon \mathcal{A}^{op}\times \mathcal{A} \to Ab$ is additive (and in both arguments separately).
For $\mathcal{A} = R$Mod and $N \in \mathcal{A}$, the functor that forms tensor product of modules $(-)\otimes N \colon \mathcal{A} \to \mathcal{A}$.
In fact thes examples are generic, see prop. 2 below.
An additive category canonically carries the structure of an Ab-enriched category where the $Ab$-enrichment structure is induced from the biproducts as described at biproduct.
With respect to the canonical Ab-enriched category-structure on additive categories $\mathcal{A}$, $\mathcal{B}$, additive functors $F : \mathcal{A} \to \mathcal{B}$ are equivalently Ab-enriched functors.
An $Ab$-enriched functor preserves all finite biproducts that exist, since finite biproducts in Ab-enriched categories are Cauchy colimits.
Let $R, R'$ be rings.
The following is the Eilenberg-Watts theorem. See there for more.
If an additive functor $F : R$Mod $\to R'$Mod is a right exact functor, then there exists an $R'$-$R$-bimodule $B$ and a natural isomorphism
with the functor that forms the tensor product with $B$.
This is (Watts, theorem 1),
In the context of derived functors in homological algebra one considers functors that are additive and in addition left/right exact functors, as discussed above in Characterization by exactness.