\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{regular monomorphism}

\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_effective_monomorphisms}{Relation to effective monomorphisms}\dotfill \pageref*{relation_to_effective_monomorphisms} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{Infty1Version}{In an $(\infty,1)$-category}\dotfill \pageref*{Infty1Version} \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 [[monomorphism]] is \emph{regular} if it behaves like an [[embedding]].

\begin{itemize}%
\item [[effective epimorphism]] $\Rightarrow$ [[regular epimorphism]] $\Leftrightarrow$ [[covering]]


\item [[effective monomorphism]] $\Rightarrow$ [[regular monomorphism]] $\Leftrightarrow$ [[embedding]] .



\end{itemize}
The universal factorization through an embedding is the [[image]].

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

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{regular monomorphism} is a [[morphism]] $f : c \to d$ in some [[category]] which occurs as the [[equalizer]] of \emph{some} [[pair of parallel morphisms]] $d \stackrel{\to}{\to} e$, i.e. for which a [[limit]] [[diagram]] of the form

\begin{displaymath}
c \stackrel{f}{\to} d \stackrel{\longrightarrow}{\longrightarrow} e
\end{displaymath}
exists.

\end{defn}
From the defining [[universal property]] of the [[limit]] it follows directly that a regular monomorphism is in particular a [[monomorphism]].

The dual concept is that of a [[regular epimorphism]].

$\backslash$begin\{rmk\} Beware that (\hyperlink{CassidyHebertKelly}{CassidyHebertKelly}) use `regular monomorphism' in a more general way: for them, a regular monomorphism is by definition the joint equalizer of an arbitrary family of parallel pairs of morphisms with common domain. This concept is sometimes called [[strict monomorphism]], dual to the more commonly used [[strict epimorphism]]. $\backslash$end\{rmk\}

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

\hypertarget{relation_to_effective_monomorphisms}{}\subsubsection*{{Relation to effective monomorphisms}}\label{relation_to_effective_monomorphisms}

\begin{defn}
\label{EffectiveMonomorphism}\hypertarget{EffectiveMonomorphism}{}
A monomorphism $i: A \to B$ is an [[effective monomorphism]] if it is the [[equalizer]] of its [[cokernel pair]]: if the [[pushout]]

\begin{displaymath}
\itexarray{
A & \stackrel{i}{\to} & B \\
i \downarrow & & \downarrow i_1 \\
B & \underset{i_2}{\to} & B +_A B
}
\end{displaymath}
exists and $i$ is the equalizer of the pair of coprojections $i_1, i_2: B \stackrel{\to}{\to} B +_A B$. Obviously effective monomorphisms are regular.

\end{defn}
\begin{prop}
\label{RegEquEff}\hypertarget{RegEquEff}{}
In a [[category]] with [[equalizers]] and [[cokernel pairs]], the class of regular monomorphism coincides with that of [[effective monomorphism]] (def. \ref{EffectiveMonomorphism}).

\end{prop}
\begin{proof}
It is clear that every effective monomorphism is regular, we need to show the converse.

Suppose $i \colon A \to B$ is the equalizer of a pair of morphisms $f, g: B \to C$, and with notation as in def. \ref{EffectiveMonomorphism}, let $j: E \to B$ be the equalizer of the pair of coprojections $i_1, i_2$. Since $f \circ i = g \circ i$, there exists a unique map $\phi: B +_A B \to C$ such that $\phi \circ i_1 = f$ and $\phi \circ i_2 = g$. Then, since

\begin{displaymath}
f j = \phi i_1 j = \phi i_2 j = g j
\end{displaymath}
and since $i: A \to B$ is the equalizer of the pair $(f, g)$, there is a unique map $k: E \to A$ such that $j = i k$. Since $i_1 i = i_2 i$, there is a unique map $l: A \to E$ such that $i = j l$. The maps $k$, $l$ are mutually inverse.

\end{proof}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{itemize}%
\item In [[Set]], or more generally in any [[pretopos]], every [[monomorphism]] is regular.


\item Similarly, in [[Ab]], and more generally any [[abelian category]], every monomorphism is regular.



\end{itemize}
\begin{prop}
\label{RegularMonomorphismsOfTopologicalSpaces}\hypertarget{RegularMonomorphismsOfTopologicalSpaces}{}
\textbf{(regular monomorphisms of [[topological spaces]])}

In the [[category]] [[Top]] of [[topological space]],

\begin{enumerate}%
\item the [[monomorphisms]] are the those [[continuous functions]] which are [[injective functions]];


\item the regular monomorphisms are the [[topological embeddings]] (that is, the injective continuous functions whose sources have the [[induced topology|topologies induced]] from their targets); these are in fact all of the [[extremal monomorphisms]].



\end{enumerate}
\end{prop}
\begin{proof}
Regarding the first statement: an injective continuous function $f \colon X \to Y$ clearly has the cancellation property that defines monomorphisms: for parallel continuous functions $g_1,g_2 \colon Z \to X$: if $f \circ g_1 = f \circ g_1$, then $g_1 = g_2$ because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if $f$ has the cacellation property, then testing on points $g_1, g_2 \colon \ast \to X$ gives that $f$ is injective.

Regarding the second statement: from the construction of [[equalizers]] in [[Top]] (\href{Top#EqualizerInTop}{this example}) we have that these are topological subspace inclusions.

Conversely, let $i \colon X \to Y$ be a [[topological subspace embedding]]. We need to show that this is the equalizer of some pair of parallel morphisms.

To that end, form the [[cokernel pair]] $(i_1, i_2)$ by taking the [[pushout]] of $i$ against itself (in the category of sets, and using the [[quotient topology]] on a [[disjoint union space]]). By prop. \ref{RegEquEff}, the equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the [[subspace topology]]. Since monomorphisms in [[Set]] are regular, we get the function $i$ back and (again by \href{Top#EqualizerInTop}{this example}) it is equipped with the subspace topology.

\end{proof}
\begin{prop}
\label{reg}\hypertarget{reg}{}
In [[Grp]], the monics are (up to [[isomorphism]]) the inclusions of [[subgroup]]s, and every monomorphism is regular.

\end{prop}
In contrast, the [[normal monomorphisms]] (where one of the morphisms $d \to e$ is required to be the [[zero morphism]]) are the inclusions of [[normal subgroups]].

\begin{proof}
The elementary proof we give follows \href{http://katmat.math.uni-bremen.de/acc/acc.pdf#page=129}{exercise 7H} of (\hyperlink{AdamekHerrlichStrecker}{AdamekHerrlichStrecker}). It is however nonconstructive (because it contains if-then-else lines); for a constructive proof, see \href{/nlab/show/Grp#eq}{here}.

Let $K \hookrightarrow H$ be a subgroup. We need to define another group $G$ and group homomorphisms $f_1, f_2 : H \to G$ such that

\begin{displaymath}
K = \{h \in H  | f_1(h) = f_2(h)\}
  \,.
\end{displaymath}
To that end, let

\begin{displaymath}
X  := H/K  \coprod  \{\hat K\}  := \{ h K | h \in H \}  \coprod  \{\hat K\}
\end{displaymath}
be the set of [[coset]]s together with one more element $\hat K$.

Let then

\begin{displaymath}
G = Aut_{Set}(X)
\end{displaymath}
be the [[permutation group]] on $X$.

Define $\rho \in G$ to be the permutation that exchanges the coset $e K$ with the extra element $\hat K$ and is the identity on all other elements.

Finally define group homomorphism $f_1,f_2 : H \to G$ by

\begin{displaymath}
f_1(h) : x \mapsto 
  \left\{
     \itexarray{
        h h' K & if x = h' K
        \\
        \hat K & if x = \hat K
     }
  \right.
\end{displaymath}
and

\begin{displaymath}
f_2(h) = \rho \circ f_1(h) \circ \rho^{-1}
  \,.
\end{displaymath}
It is clear that these maps are indeed group homomorphisms.

So for $h \in H$ we have that

\begin{displaymath}
f_1(h) : \hat K \mapsto \hat K
  \,,
\end{displaymath}
and

\begin{displaymath}
f_1(h) : e K \mapsto h K
\end{displaymath}
and

\begin{displaymath}
f_2(h) 
  : 
  \hat K 
   \mapsto 
  e K 
    \mapsto 
   h K 
   \mapsto 
  \left\{
    \itexarray{
     \hat K & if h \in K
     \\
     h K & otherwise
   }
   \right.
  \,.
\end{displaymath}
\begin{displaymath}
f_2(h) : e K \mapsto \hat K \mapsto \hat K \mapsto e K
  \,.
\end{displaymath}
So we have $f_1(h) = f_2(h)$ precisely if $h \in K$.

\end{proof}
\hypertarget{Infty1Version}{}\subsection*{{In an $(\infty,1)$-category}}\label{Infty1Version}

In the context of [[higher category theory]] the ordinary [[limit]] diagram $c \stackrel{f}{\to} d \stackrel{\to}{\to} e$ may be thought of as the beginning of a [[homotopy limit]] diagram over a [[cosimplicial object|cosimplicial]] diagram

\begin{displaymath}
c \stackrel{f}{\to} d_0 \stackrel{\to}{\to} d_1
  \stackrel{\to}{\stackrel{\to}{\to}}
  d_2
  \cdots
  \,.
\end{displaymath}
Accordingly, it is not unreasonable to define a [[regular monomorphism in an (∞,1)-category]], to be a morphism which is the [[limit in a quasi-category]] of a cosimplicial diagram.

In practice this is of particular relevance for the $\infty$-version of [[regular epimorphism]]s: with the analogous definition as described there, a morphism $f : c \to d$ is a [[regular epimorphism]] in an [[(∞,1)-category]] $C$ if for all objects $e \in C$ the induced morphism $f^* : C(d,e) \to C(c,e)$ is a [[regular monomorphism]] in [[∞Grpd]] (for instance [[model structure on simplicial sets|modeled]] by a [[homotopy limit]] over a cosimplicial diagram in [[SSet]]).

\textbf{Warning.} The same warning as at [[regular epimorphism]] applies: with this definition of regular monomorphism in an [[(∞,1)-category]] these may fail to satisfy various definitions of plain monomorphisms that one might think of.

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

\begin{itemize}%
\item [[regular epimorphism]] (containing more results which of course have duals that could be added here)

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

\begin{itemize}%
\item [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], \emph{Abstract and concrete categories: the joy of cats}. (\href{http://katmat.math.uni-bremen.de/acc/acc.pdf}{pdf})


\item [[C. Cassidy]], [[M. Hébert]], and [[Max Kelly]] \emph{Reflective subcategories, localizations and factorizationa systems}. Journal of the Australian Mathematical Society (Series A) 38.03 (1985): 287-329.



\end{itemize}
[[!redirects regular monomorphism]] [[!redirects regular monomorphisms]] [[!redirects regular mono]] [[!redirects regular monos]] [[!redirects regular monic]] [[!redirects regular monics]]

[[!redirects regular subobject]] [[!redirects regular subobjects]]



\end{document}