(also nonabelian homological algebra)
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.
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.
A category $C$ is semi-abelian if it
is Barr-exact (hence regular and in particular has finite limits);
has a zero object;
has finite coproducts; and
is protomodular.
In other words, it is a homological category which is Barr-exact and has finite coproducts.
Equivalently, $C$ is semi-abelian if:
it has finite products and coproducts and a zero object;
it has pullbacks of monomorphisms (or even only of split monomorphisms);
it has coequalizers of kernel pairs;
regular epimorphisms are stable under pullback;
equivalence relations are effective; and
the Split Short Five Lemma holds:
(split short five lemma)
Given a commutative diagram
where
$p$ and $q$ are split epimorphisms
and $l$ and $k$ are their kernels,
then if $u$ and $v$ are isomorphisms so is $w$.
To see that the second list of axioms implies the existence of finite limits, observe that the pullback
can be computed as the pullback
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.
Every abelian category is semi-abelian. Conversely, a semi-abelian category is abelian if and only if it is additive (since any exact additive category is abelian), and if and only if its opposite is semi-abelian.
The category Grp of not-necessarily-abelian groups is semi-abelian but not abelian. So are the categories of rings without units, algebras without units, Lie algebras, and many other sorts of algebras. (The category of rings with unit is not semi-abelian since it lacks a zero object.)
More generally, the category of internal group objects in any exact category is semi-abelian as soon as it has finite coproducts. For instance, this applies to internal groups in any topos with a NNO.
The opposite of any topos, such as $Set^{op}$, is Barr-exact and protomodular, but obviously lacks a zero object.
The category of Heyting semilattices
The category of (ordinary) Lie algebras
The category $Set_*$ of pointed sets is Barr-exact with finite coproducts and a zero object, but is not semi-abelian: protomodularity and the Split Short Five Lemma fail to hold.
If $C$ is exact and protomodular with finite colimits, then for any $x\in C$ the over-under category $(x/C/x)$ is semi-abelian. For example, the opposite of the category of pointed objects in a topos is semi-abelian, and in particular, $Set_*^{op}$ is semi-abelian.
Urs: how can I understand that this (has to?) involve the opposite category?
Mike: Well, as the previous example shows, $Set_*$ 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.
The categories of crossed modules, crossed complexes, and their friends are semi-abelian; see example 4.2.6 of the Van der Linden paper referenced below.
the category of cocommutative Hopf algebras over a field. Was proven in papers by Gran/Kadjo/Vercruysse and Gran/Sterck/Vercruyse
(…)
George Janelidze, László Márki, Walter Tholen, Semi-abelian categories, J. Pure Appl. Alg. 168, 2-3 (2002) 367-386, doi
Dominique Bourn, Francis Borceux, Mal'cev, protomodular, homological and semi-abelian categories, Kluwer 2004 (doi:10.1007/978-1-4020-1962-3)
Dominique Bourn, Maria Manuel Clementino, Categorical and topological aspects of semi-abelian theories , lecture notes Haute Bodeux 2007. (pdf)
Tim Van der Linden, Homology and homotopy in semi-abelian categories, math/0607100.