regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
A regular category is a finitely complete category which admits a good notion of image factorization. A primary raison d’etre behind regular categories is to have a decently behaved calculus of relations in .
A category is called regular if
We make the following remarks:
The kernel pair is always an congruence on in ; informally, is the subobject of consisting of pairs of elements which have the same value under (sometimes called the ‘kernel’ of a function in Set). The coequalizer above is supposed to be the “object of equivalence classes” of 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 is again a coequalizer diagram (nor need a regular category have coequalizers of all parallel pairs).
In fact, an equivalent definition is:
A regular category is a finitely complete category with pullback-stable image factorizations.
Here we are using “image” in the sense of “the smallest monic through which a morphism factors.” See familial regularity and exactness for a generalization of this approach to include coherent categories as well.
Examples of regular categories include the following:
Set is a regular category.
More generally, any topos is regular.
Any abelian category is regular.
If is regular, then so is the functor category for any category .
Examples of categories which are not regular include
The following example proves failure of regularity in all three cases: let be the poset ; let be the poset , and let be the poset . There is a regular epi obtained by identifying with , and there is the evident inclusion . The pullback of along is the inclusion , which is certainly an epi but not a regular epi. Hence regular epis in are not stable under pullback.
Interpreting the posets as categories, the same example works for , and also for preorders. On the other hand, the category of finite preorders is equivalent to the category finite topological spaces, so this example serves to show also that is not regular.
Let be the coequalizer of the kernel pair of . Since coequalizes its kernel pair, there is a unique map such that . It may be shown from the regular category axioms that is monic and in fact represents the image of , i.e., the smallest subobject through which factors.
A proof is spelled out on p. 32 of (vanOosten).
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.
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.
Any regular category admits a subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms: the regular coverage. If 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).
Any category with finite limits has a reg/lex completion with the following properties:
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 , or by “closing up” under images in its presheaf category ; see regular and exact completions. In general, even if is regular, is larger than (that is, it is a free cocompletion rather than merely a completion), although if satisfies the axiom of choice (in the sense that all regular epimorphisms are split), then .
Regular categories of the form for a lex category 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 can be recovered as the subcategory of projective objects. In fact, the construction of can be extended to categories having only weak finite limits, and the regular categories of the form for a “weakly lex” category 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.
Other related pages: Barr embedding theorem
Regular categories were introduced by Barr in
and by Grillet in the same volume of Lecture Notes in Mathematics.
Some of the history is provided in
A set of course notes is in section 4.1 of
An application of the regularity condition is found in the paper
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.
Enriched generalization of regular categories is considered in