κ-ary regular and exact categories
arity class: unary, finitary, infinitary
regularity
regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
exactness
exact category = unary exact
A regular category is a finitely complete category which admits a good notion of image factorization. A primary raison d’être behind regular categories $C$ is to have a decently behaved calculus of relations in $C$.
Regular categories also provide a natural semantic environment to interpret a particularly well behaved positive fragment of first order logic having connectives $\top,\wedge,\exists$ ; in other words, their internal logic is regular logic.
A category $C$ is called regular if
It is finitely complete;
the kernel pair
of any morphism $f: d \to c$ admits a coequalizer $d \times_c d \,\rightrightarrows\, d \to coeq(p_1,p_2)$;
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 $d$ in $C$; informally, $\ker(f) = d\times_c d$ 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).
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.
Even 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.
Actually, any category that is monadic over Set is regular. For example, the category of frames $Frm \simeq Loc^{op}$ is regular, and the category of compact Hausdorff spaces is regular. A proof may be found here.
Any abelian category is regular.
If $C$ is regular, then so is the functor category $C^D$ for any category $D$.
If $C$ is regular and $T$ is a Lawvere theory, then the category $Mod(T, C)$ of $T$-models in $C$ is also regular. See Theorem 5.11 in Barr’s Exact Categories.
A slice of a regular category is also regular; cf. locally regular category. So is any co-slice. (Source: Borceux-Bourn, Appendix section 5.)
If $Q$ is a quasitopos, then $Q^{op}$ is regular. Source: A2.6.3(i) in the Elephant.
Top$^{op}$ is regular. The key facts are that regular monomorphisms in $Top$ are the same as subspace inclusions, and that the pushout of a subspace inclusion is a subspace inclusion as proven here.
The category of (Hausdorff) Kelley spaces is regular (but is not, however, locally cartesian closed, nor is it exact). Source
Examples of categories which are not regular include
The following example proves failure of regularity in all three cases: let $A$ be the poset $\{a, b\} \times (0 \to 1)$; let $B$ be the poset $(0 \to 1 \to 2)$, and let $C$ be the poset $(0 \to 2)$. There is a regular epi $p: A \to B$ obtained by identifying $(a, 1)$ with $(b, 0)$, and there is the evident inclusion $i: C \to B$. The pullback of $p$ along $i$ is the inclusion $\{0, 2\} \to (0 \to 2)$, which is certainly an epi but not a regular epi. Hence regular epis in $Pos$ are not stable under pullback.
Interpreting the posets as categories, the same example works for $Cat$, and also for preorders. On the other hand, the category of finite preorders is equivalent to the category of finite topological spaces, so this example serves to show also that $Top$ is not regular.
However:
image factorization
In a regular category, every morphism $f : x\to Y$ can be factored – uniquely up to isomorphism – through its image $im(f)$ as
where $e$ is a regular epimorphism and $i$ a monomorphism.
Let $e : x \to im(f)$ be the coequalizer of the kernel pair of $f$. Since $f$ coequalizes its kernel pair, there is a unique map $i: im(f) \to c$ such that $f = i e$. 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.
A proof is spelled out on p. 32 of (vanOosten).
The classes of regular epimorphism, monomorphisms in a regular category $C$ form a factorization system.
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.
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 $C$ admits a subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms: the regular coverage. 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).
Any category $C$ with finite limits has a reg/lex completion $C_{reg/lex}$ 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 $C$, 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.
regular 2-category, regular derivator?, regular (∞,1)-category?
Regular categories were introduced in three different articles in LNM 236 by Barr, Grillet and Van Osdool, respectively:
P. A. Grillet, Regular Categories , pp.121-222.
D. H. Van Osdool, Sheaves in Regular Categories , pp.223-239.
Some of the historical context is provided in the introduction of
A nice textbook treatment can be found in chapter 2 of
More streamlined are
Peter Freyd, Andre Scedrov, Categories, Allegories , North-Holland Amsterdam 1990. (chap. 1.5. pp.68ff)
Peter Johnstone, Sketches of an Elephant I , Oxford UP 2002. (section A1.3. pp.18ff)
Dominique Bourn, Marino Gran, Regular, Protomodular, and Abelian Categories , chap. IV pp.165-211 in Pedicchio, Tholen (eds.), Categorical Foundations , Cambridge UP 2004.
A concise introductory monograph is
The following set of course notes has a section on regular categories
An application of the regularity condition^{1} is found in the paper
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
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. ↩