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 . Regular categories are also the natural setting for regular logic.
A category is regular if
It is finitely complete;
The kernel pair of any 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 of the diagram
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 ). 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.
To form the image factorization of a map , 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. 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.
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) is regular. This applies in particular to the category Ab of abelian groups.
Any abelian category is regular.
If is regular, then so is for any category .
Examples of categories which are not regular include Cat, Pos, and Top.
Further desirable exactness properties can be phrased in the language of Galois connections. For each object , consider the following relation between the class of parallel pairs ) and maps :
[to be continued]
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 . 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 .
See regular and exact completions.
H'm, so what is like if the axiom of choice fails in ? Some category like , but larger, in which every set becomes projective (but perhaps not every object in the larger category). Of course, I'd have to check that the statements above that this conclusion relies don't themselves use the axiom of choice in ! —Toby
Mike Shulman: I believe that admits the following simple description: it is the category of setoids where a morphism is a function such that implies , and two such morphisms and are set equal if for all . In this case is already exact, and therefore it is the same as . The monics in are morphisms such that implies , and the regular epics are the morphisms such that there exists a function such that for all . Evidently every set, considered as a setoid in the obvious way, is then regular projective. And of course, if satisfies AC, then every setoid is isomorphic to a set in .
Toby: This is very interesting! I should check that your construction satisfies the universal property, but for the moment I'll just believe it. Then this construction mimics the construction of sets as presets with equivalence relations. If you adopt the axiom of choice, then that construction stabilises; but otherwise, it seems that you can continue it over and over again! (Perhaps we can get a family of weaker and weaker versions of AC by postulating that the construction stabilises at some point.)
Mike Shulman: Yes, your question started me thinking in that direction too. This category of setoids is interesting in some other ways. It’s not only exact, I’m pretty sure that it’s a Π-pretopos (assuming that is). However, there’s no reason for it to be a topos even if is; in fact it may not even be well-powered (relative to )! In general the subobject lattice of a set (considered as a setoid) is the preorder reflection of . If satisfies AC, then of course any is isomorphic in this poset reflection to its image, but in the complete absence of AC I don’t see any reason for this preordered class to be small. So “ is a topos” would also be an interesting weaker form of AC.
Additionally, in the terminal object is a generator which is nondegenerate, projective, and indecomposable, but not (in general) a strong generator. Thus, it is not well-pointed in the correct sense for a pretopos. In fact, I believe saying that is a strong generator in is equivalent to AC in .
Maybe we should move this discussion to a page like exact completion of Set?.
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.
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.
Revision on November 5, 2009 at 19:15:53 by Mike Shulman See the history of this page for a list of all contributions to it.