nLab regular category

Idea

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 $C$ (see Rel).

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.

Definition

Definition

(regular category)

A category $C$ is called regular if

1. it is finitely complete;

2. the kernel pair

   $$\array{ d\times_c d &\stackrel{p_1}{\longrightarrow}& d \\ {{}^\mathllap{p_2}} \big\downarrow && \big\downarrow{{}^\mathrlap{f}} \\ d &\stackrel{f}{\longrightarrow}& c }$$

   of any morphism $f: d \to c$ admits a coequalizer $d \times_c d \,\rightrightarrows\, d \to coeq(p_1,p_2)$;

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

Remark

The kernel pair is always a 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.

Remark

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 in Def. , every regular epimorphism is the coequalizer of its kernel pair. (See for instance Lemma 5.6.6 in Practical Foundations.)

Remark

The last condition in Def. 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

$$e \;\rightrightarrows\; d \to c$$

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

In fact, a characterization equivalent to Def. is:

Proposition

A regular category is a finitely complete category with pullback-stable image factorizations in that

1. every morphism factors as a regular epimorphism followed by a monomorphism

2. these factorizations are preserved under pullback.

(eg. Gran 2020, Thm. 1.14, cf. also MO:a/252842)

See also familial regularity and exactness for a generalization of this approach to include coherent categories as well.

Examples

Examples of regular categories include the following:

Example

Set is a regular category.

Example

More generally, any topos is regular.

(Barr, Grillet & von Osdol 1971, p. 17, Borceux 1994III, Prop. 3.4.14, Johnstone 2002, p. 92)

Example

Even more generally, a locally cartesian closed category with coequalizers is regular, and so any quasitopos is regular.

Example

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.

Example

The category Grp of all groups (including non-abelian groups) is regular.

(e.g. Borceux 1994 II, Ex. 2.4.3)

Example

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.

Example

Any abelian category is regular.

Example

If $C$ is regular, then so is the functor category $C^D$ for any category $D$.

Example

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.

Example

A slice of a regular category is also regular; cf. locally regular category. So is any co-slice. (Source: Borceux-Bourn, Appendix section 5.)

Example

If $Q$ is a quasitopos, then $Q^{op}$ is regular. Source: A2.6.3(i) in the Elephant.

Example

The opposite category 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 there.

Example

The category of (Hausdorff) Kelley spaces is regular (however, is not locally cartesian closed, nor is it exact) (Cagliari-Matovani-Vitale 95)

Example

(counter-examples) Examples of categories which are not regular include

• Cat, Pos, and Top.

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 the category Top of all topological spaces is not regular.

However:

Example

(compactly generated Hausdorff spaces form a regular category)
The convenient full subcategories

$$kTop \xhookrightarrow{\;} kHaus \xhookrightarrow{\;} Top$$

of compactly generated Hausdorff spaces and of compactly generated weakly Hausdorff spaces are regular (Cagliari, Matovani & Vitale 1995, p. 3).

Example

(compactly generated $G$-spaces form a regular category) For $G \,\in\, Grp(kTop)$ a compactly generated weak Hausdorff topological group, its category of internal actions, hence the category of CGWH-topological G-spaces $G Act(kTop)$ is a regular category.

Proof

The forgetful functor $G Act(kTop) \xrightarrow{\;} kTop$ creates all limits and colimits (this Prop.). Since regularity is entirely a condition on limits and colimits (this Def.) the statement follows from Ex. .

Example

If $T$ is a Mal'cev theory (e.g., the theory of groups), then the category $Top^T$ of $T$-models in Top is regular. This is because for $T$ Mal’cev, coequalizer maps in $Top^T$ are necessarily open surjections, and open surjections are stable under pullback.

Properties

Epimorphisms

Proposition

For a morphism $f$ in a regular category, the following conditions are all equivalent:

1. $f$ is an effective epimorphism;

2. $f$ is a regular epimorphism;

3. $f$ is a strong epimorphism;

4. $f$ is an extremal epimorphism.

Pullback and pasting law

In a regular category, the usual pasting law implication also applies along regular epimorphisms also in the reverse direction, :

Proposition

In a regular category, consider a commuting diagram of the form where

1. the left square is a pullback;

2. the bottom left morphism is an regular epimorphism.

Then right right square is a pullback iff the total rectangle is.

(e.g. Gran 2020, Lem. 1.15)

Factorization properties

Proposition

image factorization

In a regular category, every morphism $f : x\to y$ can be factored – uniquely up to isomorphism – through its coimage $coim(f)$ as

$$f : x \stackrel{e}{\to} coim(f) \stackrel{i}{\to} y \,,$$

where $e$ is a regular epimorphism and $i$ a monomorphism.

Proof

Let $e : x \to coim(f)$ be the coequalizer of the kernel pair of $f$. Since $f$ coequalizes its kernel pair, there is a unique map $i: coim(f) \to y$ such that $f = i e$. It may be shown from the regular category axioms that $i$ is monic and in fact represents the coimage of $f$, i.e., the smallest subobject through which $f$ factors.

This is the mere definition of first isomorphism theorem.

A proof is spelled out on p. 30 of (vanOosten).

Proposition

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

Embedding properties

Remark

If a regular category is small, it admits particularly nice embeddings into presheaf categories. See Barr embedding theorem for more.

Axiomatizability properties

Roughly speaking, regular categories tend to be relatively well-behaved when it comes to desribing them in formalized logics.

Regular functors over a small regular category

Proposition

(Makkai)
If a regular category $\mathcal{R}$ is small, then the full subcategory of the functor category $[\mathcal{R},\mathsf{Set}]$ consisting of the regular functors only is an elementary class w.r.t. the signature given by (the underlying graph) of $\mathcal{R}$.

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 $C$ 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).

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 entries

• exact category, pretopos

• regular 2-category, regular derivator?, regular (∞,1)-category

• Barr embedding theorem

• allegory

• Rel

category	functor	internal logic	theory	hyperdoctrine	subobject poset	coverage	classifying topos
finitely complete category	cartesian functor	cartesian logic	essentially algebraic theory
lextensive category		disjunctive logic
regular category	regular functor	regular logic	regular theory	regular hyperdoctrine	infimum-semilattice	regular coverage	regular topos
coherent category	coherent functor	coherent logic	coherent theory	coherent hyperdoctrine	distributive lattice	coherent coverage	coherent topos
geometric category	geometric functor	geometric logic	geometric theory	geometric hyperdoctrine	frame	geometric coverage	Grothendieck topos
Heyting category	Heyting functor	intuitionistic first-order logic	intuitionistic first-order theory	first-order hyperdoctrine	Heyting algebra
De Morgan Heyting category		intuitionistic first-order logic with weak excluded middle			De Morgan Heyting algebra
Boolean category		classical first-order logic	classical first-order theory	Boolean hyperdoctrine	Boolean algebra
star-autonomous category		multiplicative classical linear logic
symmetric monoidal closed category		multiplicative intuitionistic linear logic
cartesian monoidal category		fragment $\{\&, \top\}$ of linear logic
cocartesian monoidal category		fragment $\{\oplus, 0\}$ of linear logic
cartesian closed category		simply typed lambda calculus

References

The notion of regular categories was introduced in:

• Michael Barr, Pierre Grillet, Donovan van Osdol, Exact Categories and Categories of Sheaves, Lec. Notes in Math. 236, Springer 1971, (doi:10.1007/BFb0058579)

Historical context:

• Peter Johnstone, Introduction of: Topos Theory (1977)

Introduction:

• Carsten Butz, Regular Categories and Regular Logic , BRICS LS-98-2 Aarhus 1998. (brics)

• Marino Gran, An introduction to regular categories, in: New Perspectives in Algebra, Topology and Categories, Coimbra Mathematical Texts, Springer (2021) [arXiv:2004.08964, ISBN:978-3-030-84319-9]

Textbook accounts:

• Peter Freyd, Andre Scedrov, Chapter 1.5 of: Categories, Allegories , North-Holland Amsterdam 1990. (chap. 1.5. pp.68ff)

• Francis Borceux, Chapter 2 of: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]

• Francis Borceux, Section 3.3.4 of: Handbook of Categorical Algebra. Vol. 3. Categories of Sheaves, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525872]

• Peter Johnstone, Section A1.3. pp.18ff of Sketches of an Elephant I , Oxford UP 2002.

• Dominique Bourn, Marino Gran, Regular, Protomodular, and Abelian Categories, Chapter IV, pp.165-211 in: Maria Pedicchio, Walter Tholen (eds.), Categorical Foundations, Cambridge University Press 2004 (doi:10.1017/CBO9781107340985.007)

The following set of course notes has a section on regular categories

• Jaap van Oosten, Basic category theory , BRICS LS-95-1 Aarhus 1995. (Section 4.1) (pdf)

An application of the regularity condition¹ is found in:

• F. Knop, Tensor envelopes of regular categories, (arXiv:math/0610552v2)

Generalization of the notion of regular categories to enriched category theory:

• Brian Day, Ross 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

• Stephen Lack, Giacomo Tendas, Enriched regular theories, J. Pure Appl. Alg. 226, n. 6 (2020) 106268, arXiv:1907.02301, doi

Regularity of (Hausdorff) compactly generated topological spaces:

• F. Cagliari, S. Mantovani, Enrico Vitale, Regularity of the category of Kelley spaces, Applied Categorical Structures volume 3, pages 357–361 (1995) (doi:10.1007/BF00872904, pdf)

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1. 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. ↩