nLab
regular category

Contents

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 CC is to have a decently behaved calculus of relations in CC.

Regular categories are also the natural setting for regular logic. For more on this see logic of regular categories.

Definition

Definition

A category CC is called regular if

  1. It is finitely complete;

  2. the kernel pair

    d× cd p 1 d p 2 f d f c \array{ d\times_c d &\stackrel{p_1}{\to}& d \\ {}^{\mathllap{p_2}}\downarrow && \downarrow^{\mathrlap{f}} \\ d &\stackrel{f}{\to}& c }

    of any morphism f:dcf: d \to c admits a coequalizer d× cddcoeq(p 1,p 2)d \times_c d \,\rightrightarrows\, d \to coeq(p_1,p_2);

  3. the pullback of a regular epimorphism along any morphism is again a regular epimorphism.

We make the following remarks:

  • The kernel pair is always an congruence on dd in CC; informally, ker(f)=d× cd\ker(f) = d\times_c d is the subobject of d×dd \times d consisting of pairs of elements which have the same value under ff (sometimes called the ‘kernel’ of a function in Set). The coequalizer above is supposed to be the “object of equivalence classes” of ker(f)\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

    edce \;\rightrightarrows\; d \to c

    along a morphism ccc' \to c is again a coequalizer diagram (nor need a regular category have coequalizers of all parallel pairs).

In fact, an equivalent definition is:

Definition

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

Examples of regular categories include the following:

Examples of categories which are not regular include

The following example proves failure of regularity in all three cases: let AA be the poset {a,b}×(01)\{a, b\} \times (0 \to 1); let BB be the poset (012)(0 \to 1 \to 2), and let CC be the poset (02)(0 \to 2). There is a regular epi p:ABp: A \to B obtained by identifying (a,1)(a, 1) with (b,0)(b, 0), and there is the evident inclusion i:CBi: C \to B. The pullback of pp along ii is the inclusion {0,2}(02)\{0, 2\} \to (0 \to 2), which is certainly an epi but not a regular epi. Hence regular epis in PosPos are not stable under pullback.

Interpreting the posets as categories, the same example works for CatCat, 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 TopTop is not regular.

Properties

Proposition

image factorization

In a regular category, every morphism f:xYf : x\to Y can be factored – uniquely up to isomorphism – through its image im(f)im(f) as

f:xeim(f)iy, f : x \stackrel{e}{\to} im(f) \stackrel{i}{\to} y \,,

where ee is a regular epimorphism and ii a monomorphism.

Proof

Let e:xim(f)e : x \to im(f) be the coequalizer of the kernel pair of ff. Since ff coequalizes its kernel pair, there is a unique map i:im(f)ci: im(f) \to c such that f=ief = i e. It may be shown from the regular category axioms that ii is monic and in fact represents the image of ff, i.e., the smallest subobject through which ff factors.

A proof is spelled out on p. 32 of (vanOosten).

Proposition

The classes of regular epimorphism, monomorphisms in a regular category CC form a factorization system.

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, as a special case of the notion of generalized kernels.

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 CC admits a subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms: the regular coverage. If CC 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 CC with finite limits has a reg/lex completion C reg/lexC_{reg/lex} with the following properties:

  • There is a full and faithful functor CC reg/lexC\hookrightarrow C_{reg/lex}
  • Each object of CC becomes projective in C reg/lexC_{reg/lex}
  • Each left-exact functor CDC\to D, where DD is regular, extends to an essentially unique regular functor C reg/lexDC_{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 CC, or by “closing up” CC under images in its presheaf category [C op,Set][C^{op},Set]; see regular and exact completions. In general, even if CC is regular, C reg/lexC_{reg/lex} is larger than CC (that is, it is a free cocompletion rather than merely a completion), although if CC satisfies the axiom of choice (in the sense that all regular epimorphisms are split), then CC reg/lexC\simeq C_{reg/lex}.

Regular categories of the form C reg/lexC_{reg/lex} for a lex category CC 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 CC can be recovered as the subcategory of projective objects. In fact, the construction of C reg/lexC_{reg/lex} can be extended to categories having only weak finite limits, and the regular categories of the form C reg/lexC_{reg/lex} for a “weakly lex” category CC 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 classes of 1-categories: exact category, coherent category, pretopos

Related classes of higher categories: regular 2-category, regular derivator?, regular (∞,1)-category?

Other related pages: Barr embedding theorem

Regular categories were introduced by Barr in

  • Michael Barr, Exact categories, Lec. Notes in Math. 236, Springer-Verlag 1971, 1-119

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

  • B. Day, R. Street, Localisation of locally presentable categories, J. Pure and Appl. Algebra 58 (1989) 227-233.

  • Dimitri Chikhladze, Barr’s embedding theorem for enriched categories, J. Pure Appl. Alg. 215, n. 9 (2011) 2148-2153, arxiv/0903.1173, doi

Revised on October 25, 2012 22:12:16 by Urs Schreiber (82.169.65.155)