nLab
semi-abelian category

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Semiabelian categories

Idea

The notion of semi-abelian category is supposed to capture the properties of categories such as that of groups, rings without unit, associative algebras without unit, Lie algebras, etc.; in generalization of how the notion of abelian categories captures the properties of the categories of abelian groups and of modules, etc.

Remark

Here it is important to consider rings and algebras without unit (really: not necessarily having a unit), since otherwise there is no zero object, and also to allow ideals to appear as subrings-without-unit.

Note that the category of rings with unit is still protomodular.

Definition

A category CC is semi-abelian if it

In other words, it is a homological category which is Barr-exact and has finite coproducts.

Equivalently, CC is semi-abelian if:

Definition

(split short five lemma)

Given a commutative diagram

L l F q C u w v K k E p B \array{ L & \overset{l}{\to} & F & \overset{q}{\to} & C \\ {}^{\mathllap{u}}\downarrow && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{v}} \\ K & \underset{k}{\to} & E& \underset{p}{\to} & B }

where

then if uu and vv are isomorphisms so is ww.

To see that the second list of axioms implies the existence of finite limits, observe that the pullback

P A f B g C\array{P & \to & A\\ \downarrow && \downarrow^f\\ B& \underset{g}{\to} & C}

can be computed as the pullback

P A×B (1,1,f) A×B (1,1,g) A×B×C\array{P & \to & A\times B\\ \downarrow && \downarrow^{(1,1,f)}\\ A\times B& \underset{(1,1,g)}{\to} & A\times B\times C}

in which both legs are split monics. Filling in one of the equivalent definitions of Barr-exactness, the equivalence of the two lists of axioms reduces to showing that in a Barr-exact category with coproducts and a zero object, protomodularity is equivalent to the Split Short Five Lemma; see the paper referenced below for a proof.

Examples

Urs: how can I understand that this (has to?) involve the opposite category?

Mike: Well, as the previous example shows, Set *Set_* itself is not semi-abelian. The way I’m thinking of it is that a surjection of pointed sets is not determined by its kernel, but an injection of pointed sets is determined by its cokernel.

Properties

General

(…)

Dold–Kan correspondence

References