Definition (-enriched category):
An -enriched category is a category such that:
- , is an abelian group.
- , is bilinear.
Definition (zero object):
A zero object is an object in an -enriched category that is both initial and terminal. We usually denote it by .
Given an -enriched category , a biproduct of is a tuple such that:
We usually denote by .
Definition (additive category):
An additive category is an -enriched category such that:
- There is a zero product in .
- Every has a biproduct.
Given in an -enriched category. A (co-)kernel of is a (co-)equalizer of and .
Definition (abelian category):
An abelian category is an additive category where:
- Every morphism has a kernel and cokernel.
- Every monomorphism is a kernel and every epimorphism is a cokernel.
The category of all left -modules of a ring is an abelian category.