category theory

# 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 $f$ 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 epimorphismsout 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.

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)