\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*{geometric realization}

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

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

[[!include category theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - 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{OfSimplicialSets}{Of cell complexes such as simplicial sets}\dotfill \pageref*{OfSimplicialSets} \linebreak
\noindent\hyperlink{OfSimplicialTopologicalSpaces}{Of simplicial topological spaces}\dotfill \pageref*{OfSimplicialTopologicalSpaces} \linebreak
\noindent\hyperlink{OfCohesiveInfinityGroupoids}{Of cohesive $\infty$-groupoids}\dotfill \pageref*{OfCohesiveInfinityGroupoids} \linebreak
\noindent\hyperlink{OfSimplicialObjects}{Of simplicial objects in a category}\dotfill \pageref*{OfSimplicialObjects} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{realizations_as_cw_complexes}{Realizations as CW complexes}\dotfill \pageref*{realizations_as_cw_complexes} \linebreak
\noindent\hyperlink{GeometricRealizationIsLeftExact}{Theorem: Geometric realization is left exact}\dotfill \pageref*{GeometricRealizationIsLeftExact} \linebreak
\noindent\hyperlink{geometric_realization_preserves_equalizers}{Geometric realization preserves equalizers}\dotfill \pageref*{geometric_realization_preserves_equalizers} \linebreak
\noindent\hyperlink{geometric_realization_preserves_finite_products}{Geometric realization preserves finite products}\dotfill \pageref*{geometric_realization_preserves_finite_products} \linebreak
\noindent\hyperlink{geometric_realization_preserves_fibrations}{Geometric realization preserves fibrations}\dotfill \pageref*{geometric_realization_preserves_fibrations} \linebreak
\noindent\hyperlink{induced_properties_of_the_fibrant_replacement}{Induced properties of the fibrant replacement}\dotfill \pageref*{induced_properties_of_the_fibrant_replacement} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{compatibility_with_homotopy_limits}{Compatibility with homotopy limits}\dotfill \pageref*{compatibility_with_homotopy_limits} \linebreak


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

\emph{Geometric realization} is the operation that builds from a [[simplicial set]] $X$ a [[topological space]] $|X|$ obtained by interpreting each element in $X_n$ -- each abstract $n$-simplex in $X$ -- as one copy of the standard topological $n$-simplex $\Delta^n_{Top}$ and then gluing together all these along their boundaries to a big topological space, using the information encoded in the face and degeneracy maps of $X$ on how these simplices are supposed to be stuck together. It generalises the geometric realization of [[simplicial complex]]es as described at that entry.

This is the special case of the general notion of [[nerve and realization]] that is induced from the standard [[simplicial set|cosimplicial]] [[topological space]] $[n] \mapsto \Delta^n_{Top}$. (N.B.: in this article, $[n]$ denotes the ordinal with $n+1$ elements. The corresponding contravariant representable is denoted $\Delta(-, n)$.)

In the context of [[homotopy theory]] geometric realization plays a notable role in the [[homotopy hypothesis]], where it is part of the [[Quillen equivalence]] between the [[model structure on topological spaces]] and the standard [[model structure on simplicial sets]].

The construction generalizes naturally to a map from [[simplicial object|simplicial]] [[topological space]]s to plain topological spaces. For more on that see [[geometric realization of simplicial spaces]].

The dual concept is \emph{[[totalization]]} .

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

There are various levels of generality in which the notion of (topological) geometric realization makes sense. The basic definition is

\begin{itemize}%
\item \hyperlink{OfSimplicialSets}{For cell complexes such as simplicial sets}.

\end{itemize}
A generalization of this of central importance is the

\begin{itemize}%
\item \hyperlink{OfSimplicialTopologicalSpaces}{Geometric realization of simplicial topological spaces}

\end{itemize}
Up to homotopy, this is a special case of a general notion of

\begin{itemize}%
\item \hyperlink{OfCohesiveInfinityGroupoids}{Geometric realization of cohesive ∞-groupoids}.

\end{itemize}
At the point-set level, it is also a special case of a general notion of

\begin{itemize}%
\item \hyperlink{OfSimplicialObjects}{Geometric realization of simplicial objects}.

\end{itemize}
\hypertarget{OfSimplicialSets}{}\subsubsection*{{Of cell complexes such as simplicial sets}}\label{OfSimplicialSets}

Let $S$ be one of the categories of [[geometric shapes for higher structures]], such as the [[globe category]] or the [[simplex category]] or the [[cube category]].

There is an obvious functor

$st : S \to$ [[Top]]

which sends the standard \emph{cellular} shape $[n]$ (the standard cellular [[globe]], [[simplex]] or [[cube]], respectively) to the corresponding standard \emph{topological} shape (for instance the standard $n$-simplex $st([n]) := \{ (x_1, \cdots, x_n) | x_i \leq x_{i+1}  \} \subset \mathbb{R}^{n}$ ) with the obvious induced face and boundary maps.

Using this, in cases where $Top$ can be regarded as [[enriched category|enriched]] over and [[copower|tensored]] over a base category $V$, the \textbf{geometric realization} of a [[presheaf]] $K^\bullet : S^{op} \to V$ on $S$ -- e.g., of a [[globular set]], a [[simplicial set]] or a [[cubical set]], respectively (when $V= Set$) -- is the [[topological space]] given by the [[end|coend]], [[weighted limit|weighted colimit]], or [[tensor product of functors]]

\begin{displaymath}
|K^\bullet| = \int^{[n] \in S} st([n]) \cdot K^n
  \,.
\end{displaymath}
In the case of [[simplicial sets]], see for more discussion also

\begin{itemize}%
\item [[nerve]], [[homotopy hypothesis]].

\end{itemize}
Via simplicial [[nerve]] functors geometric realization of simplicial sets induces geometric realizations of many other structures, for instance

\begin{itemize}%
\item [[geometric realization of categories]]

\end{itemize}
\hypertarget{OfSimplicialTopologicalSpaces}{}\subsubsection*{{Of simplicial topological spaces}}\label{OfSimplicialTopologicalSpaces}

See

\begin{itemize}%
\item [[geometric realization of simplicial topological spaces]]

\end{itemize}
\hypertarget{OfCohesiveInfinityGroupoids}{}\subsubsection*{{Of cohesive $\infty$-groupoids}}\label{OfCohesiveInfinityGroupoids}

Every [[cohesive (∞,1)-topos]] $\mathbf{H}$ (in fact every [[locally ∞-connected (∞,1)-topos]]) comes with its intrinsic notion of geometric realization.

The general abstract definition is at [[cohesive (∞,1)-topos]] in the section .

For the choice $\mathbf{H} =$ [[∞Grpd]] this reproduces the \hyperlink{OfSimplicialSets}{geometric realization of simplicial sets}, see at [[discrete ∞-groupoid]] the section 

For the choice $\mathbf{H} =$ [[ETop∞Grpd]] and [[Smooth∞Grpd]] this reproduces \hyperlink{OfSimplicialTopologicalSpaces}{geometric realization of simplicial topological spaces}. See the sections  and 

\hypertarget{OfSimplicialObjects}{}\subsubsection*{{Of simplicial objects in a category}}\label{OfSimplicialObjects}

Let $M$ be a [[cocomplete category|cocomplete]] [[simplicially enriched category]] with [[copowers]]. A \textbf{[[simplicial object]]} in $M$ is a functor $X:\Delta^{op}\to M$, where $\Delta$ is the [[simplex category]]. Its \textbf{geometric realization} is defined similarly to the classical case as a [[coend]]:

\begin{displaymath}
|X| = \int^{[n]\in\Delta} \Delta[n] \odot X_n
\end{displaymath}
where $\odot$ denotes the [[copower]] in $M$. This operation is a left adjoint which is even a simplicially enriched functor; see [[simplicial object]] for more details.

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

In this section we consider topological \hyperlink{OfSimplicialSets}{geometric realization of simplicial sets}, which is the best studied and perhaps most significant case.

\hypertarget{realizations_as_cw_complexes}{}\subsubsection*{{Realizations as CW complexes}}\label{realizations_as_cw_complexes}

Each ${|X|}$ is a CW complex (see lemma \ref{mono} below), and so geometric realization ${|(-)|}: Set^{\Delta^{op}} \to Top$ takes values in the full subcategory of CW complexes, and therefore in any [[convenient category of topological spaces]], for example in the category $CGHaus$ of compactly generated Hausdorff spaces. Let $Space$ be any convenient category of topological spaces, and let $i \colon Space \to Top$ denote the inclusion.

\begin{uprop}
For any simplicial set $X$, there is a natural isomorphism $i(\int^{n: \Delta} X(n) \cdot \sigma(n)) \cong {|X|}$, where the coend on the left is computed in $Space$.

\end{uprop}
This is obvious: more generally, if $F: J \to A$ is a diagram and $i: A \hookrightarrow B$ is a full replete subcategory, and if the colimit in $B$ of $i \circ F$ lands in $A$, then this is also the colimit of $F$ in $A$. (The dual statement also holds, with limits instead of colimits.)

Below, we let $R: Set^{\Delta^{op}} \to Space$ denote the geometric realization when considered as landing in $Space$.

\hypertarget{GeometricRealizationIsLeftExact}{}\subsubsection*{{Theorem: Geometric realization is left exact}}\label{GeometricRealizationIsLeftExact}

We continue to assume $Space$ is any [[convenient category of topological spaces]]. In this section we prove that geometric realization

\begin{displaymath}
R: Set^{\Delta^{op}} \to Space
\end{displaymath}
is a left [[exact functor]] in that it preserves [[finite limits]].

It is important that we use some such ``convenience'' assumption, because for example

\begin{displaymath}
{|(-)|}: Set^{\Delta^{op}} \to Top,
\end{displaymath}
valued in general [[topological spaces]], does not preserve products. (To get a correct statement, one usual procedure is to ``kelley-fy'' products by applying the coreflection $k: Haus \to CGHaus$. This gives the correct isomorphism in the case $Space = CGHaus$, where we have that ${|X \times Y|} \cong {|X|} \times_k {|Y|} \coloneqq k({|X|} \times {|Y|})$; the product on the right has been ``kelleyfied'' to the product appropriate for $CGHaus$.)

We reiterate that $R$ denotes the geometric realization functor considered as valued in a convenient category of spaces, whereas ${|(-)|}$ is geometric realization viewed as taking values in $Top$.

\begin{theorem}
\label{leftexact}\hypertarget{leftexact}{}
Let $U = \hom(1, -): Space \to Set$ be the underlying-set functor. Then the composite $U R: Set^{\Delta^{op}} \to Set$ is left exact.

\end{theorem}
\begin{proof}
As described at the nLab article on triangulation \href{triangulation#StandardAffineSimplexFunctor}{here}, the composite

\begin{displaymath}
\Delta \stackrel{\sigma}{\to} Space \stackrel{U}{\to} Set
\end{displaymath}
can be described as the functor

\begin{displaymath}
\Delta \cong FinInt^{op} \hookrightarrow Int^{op} \stackrel{Int(-, I)}{\to} Set
\end{displaymath}
where $Int$ is the category of intervals (linearly ordered sets with distinct top and bottom). Because every interval, in particular $I$, is a [[filtered colimit]] of finite intervals, and because finite intervals are finitely presentable intervals, it follows that $U \sigma \colon \Delta \to Set$ is a [[flat functor]] (a [[filtered colimit]] of representables). But on general grounds, [[copower|tensoring]] with a flat functor is left exact, which in this case means

\begin{displaymath}
U R = - \otimes_\Delta U \sigma: Set^{\Delta^{op}} \to Set
\end{displaymath}
is left exact.

\end{proof}
Obviously the preceding proof is not sensitive to whether we use $Space$ or $Top$.

\hypertarget{geometric_realization_preserves_equalizers}{}\paragraph*{{Geometric realization preserves equalizers}}\label{geometric_realization_preserves_equalizers}

\begin{lemma}
\label{mono}\hypertarget{mono}{}
If $i: X \to Y$ is a [[monomorphism]] of simplicial sets, then $R(i): R(X) \to R(Y)$ is a [[closed subspace]] inclusion, in fact a [[relative CW-complex]]. In particular, taking $X = \emptyset$, $R(Y)$ is a $CW$-complex.

\end{lemma}
\begin{proof}
Any monomorphism $i \colon X \to Y$ in $Set^{\Delta^{op}}$ can be seen as the result of iteratively adjoining nondegenerate $n$-simplices. In other words, there is a chain of inclusions $X = F(0) \hookrightarrow F(1) \hookrightarrow \ldots Y = colim_i F(i)$, where $F: \kappa \to Set^{\Delta^{op}}$ is a functor from some ordinal $\kappa = \{0 \leq 1\leq \ldots\}$ (as [[preorder]]) that preserves directed colimits, and each inclusion $F(\alpha \leq \alpha + 1): F(\alpha) \to F(\alpha + 1)$ fits into a pushout diagram

\begin{displaymath}
\itexarray{
\partial \Delta(-, n) & \to & F(\alpha) \\
\mathllap{j} \downarrow & & \downarrow \\
\Delta(-, n) & \to & F(\alpha+1)
}
\end{displaymath}
where $j$ is the inclusion. Now $R(j)$ is identifiable as the inclusion $S^{n-1} \to D^n$, and since $R$ preserves pushouts (which are calculated as they are in $Top$), we see by \href{/nlab/show/subspace+topology#pushout}{this lemma} that $R F(\alpha) \to R F(\alpha+1)$ is a closed subspace inclusion and evidently a [[relative CW-complex]]. By \href{/nlab/show/subspace+topology#transfinite}{another lemma}, it follows that $X \to Y$ is also a closed inclusion and indeed a relative CW-complex.

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
$R: Set^{\Delta^{op}} \to Space$ preserves [[equalizers]].

\end{cor}
\begin{proof}
The equalizer of a pair of maps in $Top$ is computed as the equalizer on the level of underlying sets, equipped with the subspace topology. So if

\begin{displaymath}
E \stackrel{i}{\to} X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} Y
\end{displaymath}
is an equalizer diagram in $Set^{\Delta^{op}}$, then ${|i|}$ is the equalizer of the pair ${|f|}$, ${|g|}$, because the underlying function $U({|i|})$ is the equalizer of $U({|f|})$, $U({|g|})$ on the underlying set level by the preceding theorem, and because ${|i|}$ is a (closed) subspace inclusion by lemma \ref{mono}. But this $Top$-equalizer ${{|i|}}: {{|E|}} \to {{|X|}}$ lives in the full subcategory $Space$, and therefore $R(i) = {|i|}$ is the equalizer of the pair $R(f) = {|f|}$, $R(g) = {|g|}$.

\end{proof}
As the proof indicates, that realization preserves equalizers is not at all sensitive to whether we use $Top$ or a convenient category of spaces $Space$.

\hypertarget{geometric_realization_preserves_finite_products}{}\paragraph*{{Geometric realization preserves finite products}}\label{geometric_realization_preserves_finite_products}

That geometric realization preserves products \emph{is} sensitive to whether we think of it as valued in $Top$ or in a convenient category $Space$. In particular, the proof uses [[cartesian closed category|cartesian closure]] of $Space$ in an essential way (in the form that [[finite products]] distribute over arbitrary [[colimits]]).

First, an easy result on products of simplices.

\begin{lemma}
\label{product}\hypertarget{product}{}
The realization of a product of two representables $\Delta(-, m) \times \Delta(-, n)$ is compact.

\end{lemma}
\begin{proof}
It suffices to observe that $\Delta[m] \times \Delta[n]$ has finitely many non-degenerate simplices. That is clear since non-degenerate $k$-simplices in the nerve of a poset $P$ are exactly injective order preserving maps $[k] \to P$.

\end{proof}
\begin{lemma}
\label{canonical}\hypertarget{canonical}{}
The canonical map

\begin{displaymath}
{|\Delta(-, m) \times \Delta(-, n)|} \to {|\Delta(-, m)|} \times {|\Delta(-, n)|}
\end{displaymath}
is a homeomorphism.

\end{lemma}
\begin{proof}
The canonical map is continuous, and a bijection at the underlying set level by theorem \ref{leftexact}. The codomain is the compact Hausdorff space $\sigma(m) \times \sigma(n)$, and the domain is compact by Lemma \ref{product}. But a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The key properties of $I$ needed for this subsection are (1) the fact it is compact Hausdorff, and (2) the order relation $\leq$ on the interval $I$ defines a closed subset of $I \times I$. These properties ensure that the affine $n$-simplex $\{(x_1, \ldots, x_n) \in I^n: x_1 \leq \ldots \leq x_n\}$ is itself compact Hausdorff, so that the proof of lemma \ref{canonical} goes through. The point is that in place of $I$, we can really use any interval $L$ that satisfies these properties, thus defining an $L$-based geometric realization instead of the standard ($I$-based) geometric realization being developed here.

\end{remark}
\begin{theorem}
\label{}\hypertarget{}{}
The functor $R: Set^{\Delta^{op}} \to Space$ preserves products.

\end{theorem}
\begin{proof}
The proof is purely formal. Let $X$ and $Y$ be simplicial sets. By the [[co-Yoneda lemma]], we have isomorphisms

\begin{displaymath}
X \cong \int^m X(m) \cdot \Delta(-, m)  \qquad Y \cong \int^n Y(n) \cdot \Delta(-, n)
\end{displaymath}
and so we calculate

\begin{displaymath}
\itexarray{
R(X \times Y) & \cong & R((\int^m X(m) \cdot \Delta(-, m)) \times (\int^n Y(n) \cdot \Delta(-, n))) \\
 & \cong & R(\int^m \int^n X(m) \cdot Y(n) \cdot (\Delta(-, m) \times \Delta(-, n))) \\
 & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot R(\Delta(-, m) \times \Delta(-, n)) \\
 & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (R(\Delta(-, m)) \times R(\Delta(-, n)) \\
 & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (\sigma(m) \times \sigma(n)) \\
 & \cong & (\int^m X(m) \cdot \sigma(m)) \times (\int^n Y(n) \cdot \sigma(n)) \\
 & \cong & R(X) \times R(Y)
}
\end{displaymath}
where in each of the second and penultimate lines, we twice used the fact that $- \times -$ preserves colimits in its separate arguments (i.e., the fact that the nice category $Space$ is cartesian closed), and the remaining lines used the fact that $R$ preserves colimits, and also products of representables by lemma \ref{canonical}.

\end{proof}
\begin{itemize}%
\item A slightly higher-level rendition of the proof might look like this:\begin{displaymath}
\itexarray{
R(X \times Y) & \cong & R((X \otimes_{\Delta} \hom) \times (Y \otimes_{\Delta} \hom)) \\
& \cong & R((X \times Y) \otimes_{\Delta \times \Delta} (\hom \times \hom)) \\
& \cong & (X \times Y) \otimes_{\Delta \times \Delta} R(\hom \times \hom) \\
& \cong & (X \times Y) \otimes_{\Delta \times \Delta} (R(\hom) \times R(\hom)) \\
& \cong & (X \otimes_{\Delta} R(\hom)) \times (Y \otimes_{\Delta} R(\hom)) \\
& \cong & R(X \otimes_{\Delta} \hom) \times R(Y \otimes_{\Delta} \hom) \\
& \cong & R(X) \times R(Y)
}
\end{displaymath}


\end{itemize}
\hypertarget{geometric_realization_preserves_fibrations}{}\subsubsection*{{Geometric realization preserves fibrations}}\label{geometric_realization_preserves_fibrations}

$\backslash$begin\{theorem\} The geometric realization of a [[Kan fibration]] is a [[Serre fibration]]. $\backslash$end\{theorem\} $\backslash$begin\{proof\} This is shown in \hyperlink{Quillen68}{Quillen 68}. $\backslash$end\{proof\}

This result implies that the geometric realization functor preserves all five classes of maps in a [[model category]]: [[weak equivalences]], [[cofibrations]], [[acyclic cofibrations]], [[fibrations]], and [[acyclic fibrations]].

In fact the geometric realization of a Kan fibration is even a [[Hurewicz fibration]] (at least relative to a [[convenient category of spaces]] in which it lives). This follows from the fact that [[a Serre fibration between CW-complexes is a Hurewicz fibration]]; a direct proof along the lines of Quillen's can be found in \hyperlink{FP90}{Fritch and Piccinini, Theorem 4.5.25}.

\hypertarget{induced_properties_of_the_fibrant_replacement}{}\subsubsection*{{Induced properties of the fibrant replacement}}\label{induced_properties_of_the_fibrant_replacement}

The previous two sections show that the geometric realization preserves finite limits and fibrations. Since its right adjoint, the singular complex functor $Top \to sSet$, also preserves both (much more trivially), and since all objects of $Top$ are fibrant and the adjunction is simplicially enriched, it follows that the composite $sSet \to Top \to sSet$ is a simplicially enriched [[fibrant replacement functor]] on $sSet$ that additionally preserves both finite limits and fibrations.

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

\begin{itemize}%
\item For $G$ a [[group]], $\mathbf{B}G$, its one-object [[groupoid]] obtained by [[delooping]], $N(\mathbf{B}G)$ the corresponding simplicial [[nerve]] [[Kan complex]], we have that the geometric realization\begin{displaymath}
\mathcal{B}G = |N\mathbf{B}G|
\end{displaymath}
is the [[topological space]] that is the [[classifying space]] for $G$-[[principal bundle]]s ([[covering space]]s), as long as we give $G$ the [[discrete topology]].



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{geometric realization}

\begin{itemize}%
\item [[geometric realization of categories|of categories]], [[geometric realization of simplicial topological spaces|of simplicial topological spaces]], [[geometric realization of cohesive ∞-groupoids|of cohesive ∞-groupoids]]

\end{itemize}

\item [[totalization]]


\item [[singular complex functor]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Daniel Quillen]], \emph{The geometric realization of a Kan fibration is a Serre fibration. Proc. Amer. Math. Soc. 19 1968 1499--1500. \href{https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1/S0002-9939-1968-0238322-1.pdf}{pdf}}


\item Fritsch and Piccinini, \emph{Cellular Structures in Topology}, Cambridge University Press, 1990, ISBN 0521327849



\end{itemize}
\hypertarget{compatibility_with_homotopy_limits}{}\subsubsection*{{Compatibility with homotopy limits}}\label{compatibility_with_homotopy_limits}

Discussion of sufficient conditions for homotopy geometric realization to be compatible with [[homotopy pullback]] (see also at \emph{[[geometric realization of simplicial topological spaces]]}):

\begin{itemize}%
\item D. Anderson, \emph{Fibrations and geometric realization} , Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (\href{http://projecteuclid.org/euclid.bams/1183541139}{euclid:1183541139})


\item [[Charles Rezk]], \emph{When are homotopy colimits compatible with homotopy base change?}, 2014 (\href{https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf}{pdf}, [[RezkHomotopyColimitsBaseChange.pdf:file]])


\item Edoardo Lanari, \emph{Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves} (\href{http://algant.eu/documents/theses/lanari.pdf}{pdf}, [[LanariHomotopyColimitsBaseChange.pdf:file]])

(expanded version of \hyperlink{Rezk14}{Rezk 14})



\end{itemize}
[[!redirects geometric realisation]]

[[!redirects geometric realizations]]

[[!redirects the geometric realization of a Kan fibration is a Serre fibration]] [[!redirects the geometric realization of a Kan fibration is a Hurewicz fibration]]



\end{document}