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\begin{document}

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\section*{regular epimorphism}

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

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

[[!include category theory - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_other_kinds_of_epimorpisms}{Relation to other kinds of epimorpisms}\dotfill \pageref*{relation_to_other_kinds_of_epimorpisms} \linebreak
\noindent\hyperlink{pullbacks_and_pushouts}{Pullbacks and pushouts}\dotfill \pageref*{pullbacks_and_pushouts} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{regular epimorphism} is a [[morphism]] $c\to d$ in a [[category]] which behaves in the way that a [[covering]] is expected to behave, in the sense that ``$d$ is the union of the parts of $c$, identified with each other in some specified way''.

In a category with [[pullbacks]], regular epimorphisms are the same as [[effective epimorphisms]] and [[strict epimorphisms]]. Every regular epimorphism is an [[epimorphism]], and every [[split epimorphism]] is regular, but the converses frequently fail.

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

A \textbf{regular epimorphism} is a [[morphism]] $f \colon c \to d$ (in a given [[category]]) that is the [[coequalizer]] of \emph{some} [[parallel pair]] of morphisms, i.e. if there exists \emph{some} [[colimit]] diagram of the form

\begin{displaymath}
a \;\rightrightarrows\; c \overset{f}{\to} d.
\end{displaymath}
The [[duality|dual]] concept is that of [[regular monomorphism]].

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

\hypertarget{relation_to_other_kinds_of_epimorpisms}{}\subsubsection*{{Relation to other kinds of epimorpisms}}\label{relation_to_other_kinds_of_epimorpisms}

The definition refers only to \emph{some} parallel pair, but often there is a canonical choice: the [[kernel pair]] of the morphism in question. A morphism having a kernel pair (such as any morphism in a category with [[pullback]]s) is a regular epimorphism if and only if it is the coequalizer of its kernel pair. See, for instance Lemma 5.6.6 in \emph{[[Practical Foundations]]}; this also follows from the theory of [[generalized kernels]]). A regular epimorphism with a kernel pair, or equivalently a morphism that is the coequalizer of its kernel pair, is here called an \textbf{[[effective epimorphism]]}.

Although the definition doesn't state so explicitly, it is true (and easy to prove) that any regular epimorphism is an [[epimorphism]]. In fact, every regular epimorphism is a [[strong epimorphism]], hence an [[extremal epimorphism]]. In particular, this implies that a regular epimorphism which is also a [[monomorphism]] must in fact be an [[isomorphism]].

Frequently (such as in a [[regular category]]), every strong or extremal epimorphism is regular. Moreover, in a regular category, every regular epimorphism is [[pullback-stability|stable]], and therefore a [[descent morphism]]. If the category is moreover [[exact category|exact]], or has stable [[reflexive coequalizers]], then every regular epimorphism is an [[effective descent morphism]].

On the other hand, every [[split epimorphism]] is regular, but the converse holds only rarely (it is an [[internalization|internal]] form of the [[axiom of choice]]).

\hypertarget{pullbacks_and_pushouts}{}\subsubsection*{{Pullbacks and pushouts}}\label{pullbacks_and_pushouts}

\begin{prop}
\label{StableUnderPullback}\hypertarget{StableUnderPullback}{}
In a [[regular category]], regular epimorphisms are preserved by [[pullback]].

\end{prop}
\begin{prop}
\label{PullbackSquareOfRegularEpisIsPushout}\hypertarget{PullbackSquareOfRegularEpisIsPushout}{}
In a [[regular category]], if in a [[pullback]] [[diagram]]

\begin{displaymath}
\itexarray{
     & \swarrow & \searrow
     \\
     & \searrow & \swarrow
  }
\end{displaymath}
the two bottom morphisms are regular epis (hence by prop. \ref{StableUnderPullback} all four morphisms are), then the diagram is also a [[pushout]].

\end{prop}
(e.g. \hyperlink{Johnstone}{Johnstone, section A, prop. 1.4.3})

\begin{prop}
\label{}\hypertarget{}{}
In a regular category $C$, regular epimorphisms and [[strong epimorphisms]] (= [[extremal epimorphisms]] since $C$ admits pullbacks) coincide; hence regular epimorphisms are closed under composition.

\end{prop}
(See \hyperlink{Johnstone}{Johnstone, A.1.3.4}.)

\begin{remark}
\label{}\hypertarget{}{}
Regular epimorphisms are not generally closed under composition. Paul Taylor in \hyperlink{Taylor}{Practical Foundations of Mathematics, p. 289} gives the following example in $Cat$. Let $\mathbb{2}$ be the [[interval category]]; then the coequalizer of the two object inclusions $1 \rightrightarrows \mathbb{2}$ is a regular epi $\mathbb{2} \to B\mathbb{N}$ (the codomain is the additive monoid $\mathbb{N}$ made into a 1-object category; the epi sends the non-identity arrow of $\mathbb{2}$ to $1 \in \mathbb{N}$). There is an evident regular epi $e: \mathbb{N} \to \mathbb{Z}/3$ of monoids. But the composite

\begin{displaymath}
\mathbb{2} \to B\mathbb{N} \stackrel{B e}{\to} B\mathbb{Z}/3
\end{displaymath}
is not a regular epi in $Cat$. A very similar example is found at \hyperlink{Bed}{Bednarczyk et al.}, Example 4.4.

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

\begin{itemize}%
\item In [[Set|the category of sets]], every epimorphism is regular. Thus, it can be hard to know, when generalising concepts from $\Set$ to other categories, what kind of epimorphism to use. Frequently, regular epimorphisms are a good choice. In particular, one may define a [[projective object]] (and hence the [[axiom of choice]]) using regular epimorphisms.


\item More generally, in every [[Grothendieck topos]] every epimorphism is regular (and in fact [[effective epimorphism|effective]], see there).


\item In [[Grp|the category of groups]], every epimorphism is regular. A number of proofs can be found in the literature; one proof that avoids case analysis is given \href{http://ncatlab.org/toddtrimble/published/epimorphisms+in+the+category+of+groups}{here}.


\item In the category of [[monoid]]s, the inclusion $\mathbb{N}\hookrightarrow\mathbb{Z}$ is an epimorphism, even though it is far from a [[surjection]]. But in this or any other [[algebraic category]] (a category of models of an [[algebraic theory]]), the morphisms whose underlying function is surjective are precisely the regular epimorphisms. Thus $\mathbb{N}\hookrightarrow\mathbb{Z}$ is not a regular epimorphism.


\item Similarly, in the category of rings, not every epimorphism is regular, as shown by the example $\mathbb{Z} \hookrightarrow \mathbb{Q}$.


\item In [[Diff]], the category of smooth (paracompact) manifolds, regular epimorphisms are not as useful as in other settings. As any [[split epimorphism]] is regular, and split epimorphisms are badly behaved in $\Diff$ (for example, pullbacks of split epis do not necessarily exist), the usual procedure is to consider the smallest class of arrows inside regular epis of which all pullbacks exist, namely the surjective [[submersions]]. In the setting of [[differentiable stacks]] and [[Lie groupoids]] it is surjective submersions that play the role of regular epimorphisms.



\end{itemize}
\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item The page [[epimorphism]] has a list of many types of epimorphism and their relationships.


\item The dual concept is [[regular monomorphism]]. In particular, in the presence of pullbacks, [[effective epimorphisms]] and [[strict epimorphisms]] are the same as regular ones.


\item In higher category theory:

\begin{itemize}%
\item [[regular epimorphism in an (∞,1)-category]]

\end{itemize}


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

\begin{itemize}%
\item [[Peter Johnstone]], Section A \emph{[[Sketches of an Elephant]]}


\item Marek A. Bednarczyk, Andrzej M. Borzyszkowski, and Wieslaw Pawlowski, \emph{Generalized congruences -- epimorphisms in $\mathcal{C}a t$}, Theory and Applications of Categories electronic only 5 (1999), 266-280. \href{https://eudml.org/doc/120226?lang=fr&limit=5}{https://eudml.org/doc/120226}


\item [[Paul Taylor]], \emph{[[Practical Foundations of Mathematics]]}, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999.



\end{itemize}
[[!redirects regular epimorphisms]] [[!redirects regular epi]] [[!redirects regular epis]] [[!redirects regular epic]] [[!redirects regular epics]]



\end{document}