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\section*{regular category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{regular_and_exact_categories}{}\paragraph*{{Regular and Exact categories}}\label{regular_and_exact_categories}

[[!include regular and exact categories - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category Theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{factorization_properties}{Factorization properties}\dotfill \pageref*{factorization_properties} \linebreak
\noindent\hyperlink{embedding_properties}{Embedding properties}\dotfill \pageref*{embedding_properties} \linebreak
\noindent\hyperlink{stronger_conditions}{Stronger conditions}\dotfill \pageref*{stronger_conditions} \linebreak
\noindent\hyperlink{exactness}{Exactness}\dotfill \pageref*{exactness} \linebreak
\noindent\hyperlink{higher_arity}{Higher arity}\dotfill \pageref*{higher_arity} \linebreak
\noindent\hyperlink{the_regular_topology}{The regular topology}\dotfill \pageref*{the_regular_topology} \linebreak
\noindent\hyperlink{making_categories_regular}{Making categories regular}\dotfill \pageref*{making_categories_regular} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{regular category} is a [[finitely complete category]] which admits a good notion of [[image]] factorization. A primary \emph{raison d'\^e{}tre} behind regular categories $C$ is to have a decently behaved \emph{calculus of [[relation]]s} in $C$.

Regular categories also provide a natural [[semantics|semantic]] environment to [[interpretation|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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{udefn}
A [[category]] $C$ is called \textbf{regular} if

\begin{enumerate}%
\item It is [[finitely complete category|finitely complete]];


\item the [[kernel pair]]

\begin{displaymath}
\itexarray{
    d\times_c d &\stackrel{p_1}{\to}& d
    \\
    {}^{\mathllap{p_2}}\downarrow && \downarrow^{\mathrlap{f}}
    \\
    d &\stackrel{f}{\to}& c
  }
\end{displaymath}
of any [[morphism]] $f: d \to c$ admits a [[coequalizer]] $d \times_c d \,\rightrightarrows\, d \to coeq(p_1,p_2)$;


\item the [[pullback]] of a [[regular epimorphism]] along any morphism is again a regular epimorphism.



\end{enumerate}
\end{udefn}
We make the following remarks:

\begin{itemize}%
\item 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]].


\item A map which is the coequalizer of a parallel pair of morphisms is called a \emph{[[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 \emph{[[Practical Foundations]]}.)


\item 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

\begin{displaymath}
e \;\rightrightarrows\; d \to c
\end{displaymath}
along a morphism $c' \to c$ is again a coequalizer diagram (nor need a regular category have coequalizers of all parallel pairs).



\end{itemize}
In fact, an equivalent definition is:

\begin{udefn}
A \textbf{regular category} is a finitely complete category with pullback-stable [[image]] factorizations.

\end{udefn}
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 category|coherent]] categories as well.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Examples of regular categories include the following:

\begin{itemize}%
\item [[Set]] is a regular category.


\item More generally, any [[topos]] is regular.


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


\item The category of [[algebra over a Lawvere theory|models]] of any finitary [[algebraic theory]] (i.e., [[Lawvere theory]]) $T$ is regular. This applies in particular to the category [[Ab]] of [[abelian group]]s.


\item Actually, any category that is [[monadic functor|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 \href{/nlab/show/colimits+in+categories+of+algebras#exact}{here}.


\item Any [[abelian category]] is regular.


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


\item 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 \hyperlink{Barr}{Exact Categories}.


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


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


\item [[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 \href{/nlab/show/subspace+topology#pushout}{here}.


\item The category of ([[Hausdorff space|Hausdorff]]) [[Kelley spaces]] is regular (but is not, however, locally cartesian closed, nor is it [[exact category|exact]]). \href{http://www.dm.unibo.it/~cagliari/articoli/Regularkelley.pdf}{Source}



\end{itemize}
Examples of categories which are \textbf{not regular} include

\begin{itemize}%
\item [[Cat]], [[Pos]], and [[Top]].

\end{itemize}
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:

\begin{itemize}%
\item 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 coequalizer maps in $Top^T$ are necessarily open surjections, and open surjections are stable under pullback.

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{factorization_properties}{}\subsubsection*{{Factorization properties}}\label{factorization_properties}

\begin{uprop}
\textbf{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

\begin{displaymath}
f : x \stackrel{e}{\to} im(f) \stackrel{i}{\to} y
  \,,
\end{displaymath}
where $e$ is a [[regular epimorphism]] and $i$ a [[monomorphism]].

\end{uprop}
\begin{proof}
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 (\hyperlink{vanOosten}{vanOosten}).

\end{proof}
\begin{uprop}
The classes of [[regular epimorphism]], [[monomorphism]]s in a regular category $C$ form a [[orthogonal factorization system|factorization system]].

\end{uprop}
\hypertarget{embedding_properties}{}\subsubsection*{{Embedding properties}}\label{embedding_properties}

\begin{ubarrembeddingtheorem}
If a regular category is small, it admits particularly nice embeddings into presheaf categories. See [[Barr embedding theorem]] for more.

\hypertarget{axiomatizability_properties}{}\subsubsection*{{Axiomatizability properties}}\label{axiomatizability_properties}

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

\hypertarget{regular_functors_over_a_small_regular_category}{}\paragraph*{{Regular functors over a small regular category}}\label{regular_functors_over_a_small_regular_category}

\hypertarget{a_result_of_makkai}{}\paragraph*{{A result of Makkai}}\label{a_result_of_makkai}

\hypertarget{proposition_makkai}{}\paragraph*{{Proposition (Makkai)}}\label{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 $\mathcal{R}$.

\end{ubarrembeddingtheorem}
\hypertarget{stronger_conditions}{}\subsection*{{Stronger conditions}}\label{stronger_conditions}

\hypertarget{exactness}{}\subsubsection*{{Exactness}}\label{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 category|cocomplete]] without being exact.

Regularity and exactness can also be phrased in the language of [[Galois connection]]s, as a special case of the notion of [[generalized kernels]].

\hypertarget{higher_arity}{}\subsubsection*{{Higher arity}}\label{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.

\hypertarget{the_regular_topology}{}\subsection*{{The regular topology}}\label{the_regular_topology}

Any regular category $C$ admits a [[subcanonical site|subcanonical]] [[Grothendieck topology]] whose covering families are generated by single [[regular epimorphisms]]: the [[regular coverage]]. If $C$ is [[exact category|exact]] or has pullback-stable [[reflexive coequalizer]]s, 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).

\hypertarget{making_categories_regular}{}\subsection*{{Making categories regular}}\label{making_categories_regular}

Any category $C$ with [[finite limits]] has a \textbf{reg/lex completion} $C_{reg/lex}$ with the following properties:

\begin{itemize}%
\item There is a [[full and faithful functor]] $C\hookrightarrow C_{reg/lex}$
\item Each object of $C$ becomes [[projective object|projective]] in $C_{reg/lex}$
\item Each left-exact functor $C\to D$, where $D$ is regular, extends to an essentially unique [[regular functor]] $C_{reg/lex}\to D$.

\end{itemize}
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 category|exact one]], the result is unsurprisingly the [[ex/lex completion]]. See [[regular and exact completions]] for more about all of these operations.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[exact category]], [[coherent category]], [[pretopos]]


\item [[regular 2-category]], [[regular derivator]], [[regular (∞,1)-category]]


\item [[regular logic]]


\item [[Barr embedding theorem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Regular categories were introduced in three different articles in LNM \textbf{236} by Barr, Grillet and Van Osdool, respectively:

\begin{itemize}%
\item [[Michael Barr]], \emph{Exact categories}, Lec. Notes in Math. \textbf{236}, Springer-Verlag 1971, 1-119. (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.732.4603&rep=rep1&type=pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item P. A. Grillet, \emph{Regular Categories} , pp.121-222.


\item D. H. Van Osdool, \emph{Sheaves in Regular Categories} , pp.223-239.



\end{itemize}
Some of the historical context is provided in the introduction of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Topos Theory} (1977)

\end{itemize}
A nice textbook treatment can be found in chapter 2 of

\begin{itemize}%
\item [[Francis Borceux]], \emph{Handbook of Categorical Algebra 2: Categories and Structures} , Cambridge UP 1994.

\end{itemize}
More streamlined are

\begin{itemize}%
\item [[Peter Freyd]], Andre Scedrov, \emph{Categories, Allegories} , North-Holland Amsterdam 1990. (chap. 1.5. pp.68ff)


\item [[Peter Johnstone]], \emph{Sketches of an Elephant I} , Oxford UP 2002. (section A1.3. pp.18ff)


\item [[Dominique Bourn]], [[Marino Gran]], \emph{Regular, Protomodular, and Abelian Categories} , chap. IV pp.165-211 in Pedicchio, Tholen (eds.), \emph{Categorical Foundations} , Cambridge UP 2004.



\end{itemize}
A concise introductory monograph is

\begin{itemize}%
\item [[Carsten Butz]], \emph{Regular Categories and Regular Logic} , BRICS LS-98-2 Aarhus 1998. (\href{http://www.brics.dk/LS/98/2/}{brics})

\end{itemize}
The following set of course notes has a section on regular categories

\begin{itemize}%
\item [[Jaap van Oosten]], \emph{Basic category theory} , BRICS LS-95-1 Aarhus 1995. (\href{http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf#page=30}{Section 4.1}) (\href{http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf}{pdf})

\end{itemize}
An application of the regularity condition\footnote{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.}  is found in the paper

\begin{itemize}%
\item F. Knop, \emph{Tensor envelopes of regular categories}, (\href{http://arxiv.org/abs/math/0610552}{arXiv:math/0610552v2})

\end{itemize}
Enriched generalization of regular categories is considered in

\begin{itemize}%
\item B. Day, R. Street, \emph{Localisation of locally presentable categories}, J. Pure and Appl. Algebra \textbf{58} (1989) 227-233.


\item Dimitri Chikhladze, \emph{Barr's embedding theorem for enriched categories}, J. Pure Appl. Alg. \textbf{215}, n. 9 (2011) 2148-2153, \href{http://arxiv.org/abs/0903.1173}{arxiv/0903.1173}, \href{http://dx.doi.org/10.1016/j.jpaa.2010.12.004}{doi}



\end{itemize}
[[!redirects regular categories]]



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