(also nonabelian homological algebra)
The axioms of a semi-abelian category are supposed to capture the properties of the categories of groups, rings without unit, associative algebras without unit, Lie algebras, as nicely as the axioms of an abelian category captures the properties of the categories of abelian groups and of modules.
Mike: Why only rings without units (that is, rngs)? Intuitively, what important properties do the above listed examples share that are not shared by rings with units?
Zoran Skoda: I want to know the answer as well. It might be something in the self-dual axioms. For unital rings artinian implies noetherian but not other way around; though the definitions of the two notions are dual.
Toby: The category of unital rings and unitary ring homomorphisms has no zero object.
Mike: Ah, right. Is it protomodular? I think I will understand this definition better from some non-examples that violate each clause individually.
walt: It is protomodular. This follows from the main theorem of Characterization of Protomodular Varieties of Universal Algebra by Bourn and Janelidze. By that theorem any variety that contains a group will be protomodular. Unital rings only fail to be semiabelian for the trivial reason that ideals aren’t subrings.
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.
(…)
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.
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.