Definition
An abelian category is a Category that behaves like the category of abelian groups or modules. It provides an abstract framework for homological algebra.
A category is abelian if:
- is additive
- Every morphism has a kernel and cokernel
- Every monomorphism is the kernel of its cokernel
- Every epimorphism is the cokernel of its kernel
Properties
- All finite limits and colimits exist
- The notions of monomorphism/epimorphism coincide with kernel/cokernel relationships
- Every morphism factors as an epimorphism followed by a monomorphism
- The Snake Lemma and Five Lemma hold
Examples
- Ab (Category): The category of abelian groups
- The category of modules over a ring
- The category of sheaves of abelian groups on a topological space
- The category of finite-dimensional vector spaces
Applications
Abelian categories are the natural setting for:
- Homological algebra
- Derived functors
- Spectral sequences
- Cohomology theories
Related Concepts
- Additive Category: A weaker structure
- Category: The general notion
- Ab (Category): The prototypical example
- Homological Algebra: The theory developed in abelian categories