nLab pre-abelian category

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 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 $f$ and $g$ in an Ab-enriched category is the same as a kernel of $f-g$. Dually for finite colimits, coequalizers and cokernels.

Properties

Proposition

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

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

Proposition

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

$$A\stackrel{p}\to \coker(\ker f)\stackrel{\bar{f}}\to\ker(\coker f)\stackrel{i}\to B$$

where $p$ is a cokernel, hence an epi, and $i$ is a kernel, and hence monic.

Remark

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

Related concepts

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