additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
A functor between additive categories is itself called additive if it preserves finite biproducts.
That is,
maps a zero object to a zero object, ;
given any two objects , there is an isomorphism , 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 preserves biproducts (where and are categories with a zero object):
The hom-functor is additive in both arguments separately (using the nature of biproducts and that hom-functors preserve limits in each variable separately).
For Mod and , the functor that forms tensor product of modules .
In fact these examples are generic, see prop. below.
Every solid abelian group is by definition an additive functor.
An additive category canonically carries the structure of an Ab-enriched category where the -enrichment structure is induced from the biproducts as described at biproduct.
With respect to the canonical Ab-enriched category-structure on additive categories , , additive functors are equivalently Ab-enriched functors.
An -enriched functor preserves all finite biproducts that exist, since finite biproducts in Ab-enriched categories are Cauchy colimits.
Let be rings.
The following is the Eilenberg-Watts theorem. See there for more.
If an additive functor Mod Mod is a right exact functor, then there exists an --bimodule and a natural isomorphism
with the functor that forms the tensor product with .
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.
Last revised on April 15, 2023 at 09:59:22. See the history of this page for a list of all contributions to it.