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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.

Idea

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 epimorphism along any morphism is again a regular epimorphism.

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 congruence 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 $\ker(f)$ as an internal equivalence relation.

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 Lemma 5.6.6 in Practical 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 (nor need a regular category have coequalizers of all parallel pairs).

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.

Stronger conditions

Exactness

If a regular category additionally has the property that every congruence is a kernel pair (and hence has a quotient), then it is called a (Barr-) exact category. Note that while regularity implies the existence of some coequalizers, and exactness implies the existence of more, an exact category need not have all coequalizers (only coequalizers of congruences), whereas a regular category can be cocomplete without being exact.

Regularity and exactness can also 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]

Higher arity

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. A natural generalization is to include (finite or infinite) unions of subobjects, or equivalently images of (finite or infinite) families as well as of single morphisms. This leads to the notion of coherent category.

Just as regularity implies the existence of certain coequalizers, coherence implies the existence of certain coproducts and pushouts, but not all. A lextensive category has all (finite or infinite) coproducts that are disjoint and stable under pullback. It is easy to see that a lextensive regular category must actually be coherent.

The regular topology

Any regular category $C$ admits a subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms. If $C$ is exact or has pullback-stable reflexive coequalizers, then its codomain fibration is a stack for this topology (the necessary and sufficient condition is that any pullback of a kernel pair is again a kernel pair).

Making categories regular

Any category $C$ with finite limits has a reg/lex completion $C_{reg/lex}$ with the following properties:

• There is a full and faithful functor $C\hookrightarrow C_{reg/lex}$
• Each object of $C$ becomes projective in $C_{reg/lex}$
• Each left-exact functor $C\to D$, where $D$ is regular, extends to an essentially unique regular functor $C_{reg/lex}\to D$.

In particular, the reg/lex completion is a left adjoint to the forgetful functor from regular categories to lex categories (categories with finite limits). The reg/lex completion can be obtained by “formally adding images” for all morphisms in $C$, or by “closing up” $C$ under images in its presheaf category $[C^{op},Set]$; see regular and exact completions. In general, even if $C$ is regular, $C_{reg/lex}$ is larger than $C$ (that is, it is a free cocompletion rather than merely a completion), although if $C$ satisfies the axiom of choice (in the sense that all regular epimorphisms are split), then $C\simeq C_{reg/lex}$.

Regular categories of the form $C_{reg/lex}$ for a lex category $C$ can be characterized as those regular categories in which every object admits both a regular epi from a projective object and a monomorphism into a projective object, and the projective objects are closed under finite limits. In this case $C$ can be recovered as the subcategory of projective objects. In fact, the construction of $C_{reg/lex}$ can be extended to categories having only weak finite limits, and the regular categories of the form $C_{reg/lex}$ for a “weakly lex” category $C$ are those satisfying the first two conditions but not the third.

When the reg/lex completion is followed by the ex/reg completion which completes a regular category into an exact one, the result is unsurprisingly the ex/lex completion. See regular and exact completions for more about all of these operations.

Related Articles articles

Regular categories were introduced by Barr in Exact categories, by M. Barr, Lecture Notes in Mathematics, Springer-Verlag 1971 and by Grillet in the same volume of Lecture Notes in Mathematics.

Some of the history is provided in Topos Theory, by P. Johnstone, 1977.

An application of the regularity condition is found in the paper Tensor envelopes of regular categories, by F. Knop. arXiv:math/0610552v2

Knop’s condition for regularity is slightly different from that presented here; he works with categories that when augmented by an absolutely initial object are regular in the terminology here. In the paper, Knop generalizes a construction of Deligne by showing how to construct a symmetric pseudo-abelian tensor category out of a regular category through the calculus of relations.