Enriched category theory
Could not include enriched category theory - contents
Additive and abelian categories
A functor between additive categories is itself called additive if it preserves finite biproducts.
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:
The hom-functor is additive (and in both arguments separately).
For Mod and , the functor that forms tensor product of modules .
In fact thes examples are generic, see prop. 2 below.
Relation to -enriched functors
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-structre on additive categories , , additive functors are equivalently Ab-enriched functors.
Characterization of right exact additive functors
Let be rings.
The following is the Eilenberg-Watts theorem. See there for more.
This is (Watts, theorem 1),
- Charles Watts?, Intrinsic characterizations of some additive functors, Proceedings of the American Mathematical Society (1959) (JSTOR)
Revised on March 11, 2014 02:53:28
by Urs Schreiber