Homological Algebra/Defintion of abelian category

Definition (-enriched category):

An -enriched category is a category such that:

  1. , is an abelian group.
  2. , 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 .

Definition (biproduct):

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:

  1. There is a zero product in .
  2. Every has a biproduct.

Definition ((co-)kernel):

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:

  1. Every morphism has a kernel and cokernel.
  2. Every monomorphism is a kernel and every epimorphism is a cokernel.

Example:

The category of all left -modules of a ring is an abelian category.

ExercisesEdit

  1. Given   in an  -enriched category with zero object. Prove that   iff   factors through  .
  1. Given a biproduct   of   and  . Prove that   is a coproduct of   and   and   is a product of   and  .
  1. In an  -enriched category with zero object, a kernel of   can be equivalently be characterized as a pullback of   along  .