nLab regular category (Rev #12, changes)

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A regular category is a finitely complete category which admits a good notion of image factorization. A primary raison d’etre behind regular categories $C$ is to have a decently behaved calculus of relations in $C$. Regular categories are also the natural setting for regular logic.

Definition

A category $C$ is regular if

• It is finitely complete;

• The kernel pair $p_1, p_2: e \,\rightrightarrows\, d$ of any $f: d \to c$ admits a coequalizer; and

• The pullback of a regular epi along any map is a regular epi.

Here the “kernel pair” is the parallel pair of projection maps coming out of a pullback $e$ of the diagram

The kernel pair is always an internal equivalence relation on $d$ in $C$; informally, $\ker(f)$ is the subobject of $d \times d$ consisting of pairs of elements which have the same value under $f$ (sometimes called the ‘kernel’ of a function in $\Set$). The coequalizer above is supposed to be the “object of equivalence classes” of the equivalence relation $\ker(f)$.

A map which is the coequalizer of a parallel pair of morphisms is called a regular epimorphism . In fact, in any category satisfying the first two conditions above, every coequalizer is the coequalizer of its kernel pair. (See for instance Paul Taylor’s Practical Foundations of Mathematics, Lemma 5.6.6.) 5.6.6 inPractical Foundations.)

The last condition may equivalently be stated in the form “coequalizers of kernel pairs are stable under pullback”. However, it is not generally true in a regular category that the pullback of a general coequalizer diagram

along a morphism $c' \to c$ is again a coequalizer diagram.

To form the image factorization of a map $f: d \to c$, let $q: d \to k$ be the coequalizer of the kernel pair of $f$. Since $f$ coequalizes its kernel pair, there is a unique map $i: k \to c$ such that $f = i q$. It may be shown from the regular category axioms that $i$ is monic and in fact represents the image of $f$, i.e., the smallest subobject through which $f$ factors. Moreover, the classes of regular epis and of (all) monomorphisms form a factorization system.

In fact, a regular category can alternately be defined as a finitely complete category with pullback-stable image factorizations. See familial regularity and exactness for a generalization of this approach to include coherent categories as well.

Examples

• Set is a regular category. In fact, any topos is regular. More generally, a locally cartesian closed category with coequalizers is regular, and so any quasitopos is regular.

• The category of models of any finitary algebraic theory (i.e., Lawvere theory) $T$ is regular. This applies in particular to the category Ab of abelian groups.

• Any abelian category is regular.

• If $C$ is regular, then so is $C^D$ for any category $D$.

Examples of categories which are not regular include Cat, Pos, and Top.

Remarks

1. As exactness properties go, the ones possessed by general regular categories are fairly moderate; the main condition is of course stability of regular epis under pullback. In the first place, the focus is just on certain coequalizers; finite coproducts aren’t even mentioned. Some of that imbalance is redressed by the notion of lextensive category, where coproducts are also stable under pullback (and are also disjoint).

Further desirable exactness properties can be phrased in the language of Galois connections. For each object $d$, consider the following relation $coeq$ between the class of parallel pairs $f, g: e \stackrel{\to}{\to} d$) and maps $h: d \to c$:

[to be continued]