nLab
pre-abelian category

Contents

Definition

Definition

A pre-abelian category is an additive category (an Ab-enriched category with finite biproducts) such that every morphism has a kernel and a cokernel.

Equivalently:

Definition

A pre-abelian category is an Ab-enriched category category with all finite limits and finite colimits.

Proposition

These two definitions are indeed equivalent.

Proof

By the discussion here the existence of finite limits is equivalent to that of finite products and equalizers. But an equalizer of two morphisms ff and gg in an Ab-enriched category is the same as a kernel of fgf-g. Dually for finite colimits, coequalizers and cokernels.

Properties

Proposition

For every object cCc\in C in a pre-abelian category, the operations of kernel and cokernel form a Galois connection between the preorders Sub(c)Sub(c) of monomorphisms (subobjects) into cc and Quot(c)Quot(c) of epimorphismsout of cc.

In particular, f:bcf:b\to c is a kernel iff f=ker(coker(f))f = ker(coker(f)) and dually.

Proposition

Every morphism f:ABf:A\to B in a pre-abelian category has a canonical decomposition

Apcoker(kerf)f¯ker(cokerf)iB A\stackrel{p}\to \coker(\ker f)\stackrel{\bar{f}}\to\ker(\coker f)\stackrel{i}\to B

where pp is a cokernel, hence an epi, and ii is a kernel, and hence monic.

Remark

If f¯\bar f in the above decomposition is always an isomorphism, then the pre-abelian category is called an abelian category.

Examples

  • Of course every abelian category is pre-abelian.

  • The category TFTF of torsion-free abelian groups is reflective in all of Ab. Therefore, it is a complete and cocomplete AbAb-enriched category, and therefore in particular pre-abelian. However, it is not abelian; the monomorphism 2:2:\mathbb{Z}\to \mathbb{Z} is not a kernel.

The concept “pre-abelian category” is part of a sequence of concepts of additive and abelian categories.

Revised on August 27, 2012 21:49:47 by Urs Schreiber (89.204.130.6)