nLab
semi-abelian category

Idea

The axioms of a semi-abelian category are supposed to capture the properties of the categories of groups, rings without unit, algebras without unit, Lie algebras, as nicely as the axioms of an abelian category captures the properties of the category 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.

Definition

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 Bourn-protomodular?.

Mike: I was about to create the link to Bourn-protomodular category, but then I wondered whether we should just say “protomodular”—is there a reason to disambiguate it with the prefix “Bourn-“? We (sometimes) say “Barr-exact” to avoid confusion with Quillen-exact; is there any ambiguity for “protomodular?”

Zoran Skoda: I think not, I saw just one notion of protomodular so far (unlike word semi-abelian which has been used for some other things, mainly before the now accepted dominant notion); besides the notion is mainly used by one and the same community (unlike word exact category which lives with different meanings in completely different societies of mathematicians). I think no need to call it Bourn-protomodular; of course the entry should have pointers to the original references and short mention of the optional modifier, I think.

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: given a commutative diagram

  $$\begin{array}{ccccc}L& \stackrel{l}{\to }& F& \stackrel{q}{\to }& C\\ {}^u\downarrow & & {\downarrow }^w& & {\downarrow }^v\\ K& \underset{k}{\to }& E& \underset{p}{\to }& B\end{array}$$\array{L & \overset{l}{\to} & F & \overset{q}{\to} & C\\^u\downarrow && \downarrow^w && \downarrow^v \\ K & \underset{k}{\to} & E& \underset{p}{\to} & B}

  where $p$ and $q$ are split epimorphisms and $l$ and $k$ are their kernels, if $u$ and $v$ are isomorphisms then so is $w$.

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

$$\begin{array}{ccc}P& \to & A\\ \downarrow & & {\downarrow }^f\\ B& \underset{g}{\to }& C\end{array}$$\array{P & \to & A\\ \downarrow && \downarrow^f\\ B& \underset{g}{\to} & C}

can be computed as the pullback

$$\begin{array}{ccc}P& \to & A\times B\\ \downarrow & & {\downarrow }^{(\mathrm{1,1},f)}\\ A\times B& \underset{(\mathrm{1,1},g)}{\to }& A\times B\times C\end{array}$$\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.

Remarks

• In its most general form, the Dold-Kan correspondence holds for simplicial objects in semi-abelian categories.

Examples

• 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 ${\mathrm{Set}}^{\mathrm{op}}$, is Barr-exact and protomodular, but obviously lacks a zero object.

• The category of Heyting semilattices

• The category of (ordinary) Lie algebras

• The category ${\mathrm{Set}}_{*}$ 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, ${\mathrm{Set}}_{*}^{\mathrm{op}}$ is semi-abelian.

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

Mike: Well, as the previous example shows, ${\mathrm{Set}}_{*}$ 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 2.6(4) of the paper referenced below.

References

• George Janelidze, Lászlóe Márki, Walter Tholen, Semi-Abelian Categories (web)

• Dominique Bourn, Francis Borceux, Mal’cev, protomodular, homological and semi-abelian categories, Kluwer 2004.