nLab pre-abelian category





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.



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


These two definitions are indeed equivalent.


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.



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 epimorphisms out of cc.

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


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.


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


  • 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.

Last revised on February 28, 2024 at 18:25:33. See the history of this page for a list of all contributions to it.