\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 of simplicial topological spaces}

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

\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{reminder_on_nice_simplicial_topological_spaces}{Reminder on nice simplicial topological spaces}\dotfill \pageref*{reminder_on_nice_simplicial_topological_spaces} \linebreak
\noindent\hyperlink{bisimplicial_sets_and_good_resolutions}{Bisimplicial sets and good resolutions}\dotfill \pageref*{bisimplicial_sets_and_good_resolutions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_between_fat_and_ordinary_geometric_realization}{Relation between fat and ordinary geometric realization}\dotfill \pageref*{relation_between_fat_and_ordinary_geometric_realization} \linebreak
\noindent\hyperlink{compatibility_with_limits}{Compatibility with limits}\dotfill \pageref*{compatibility_with_limits} \linebreak
\noindent\hyperlink{ordinary_geometric_realization}{Ordinary geometric realization}\dotfill \pageref*{ordinary_geometric_realization} \linebreak
\noindent\hyperlink{fat_geometric_realization}{Fat geometric realization}\dotfill \pageref*{fat_geometric_realization} \linebreak
\noindent\hyperlink{RelationToHomotopyColimit}{Relation to the homotopy colimit}\dotfill \pageref*{RelationToHomotopyColimit} \linebreak
\noindent\hyperlink{examples_and_applications}{Examples and Applications}\dotfill \pageref*{examples_and_applications} \linebreak
\noindent\hyperlink{TopologicalPrincipalInfinityBundle}{Topological principal $\infty$-bundles}\dotfill \pageref*{TopologicalPrincipalInfinityBundle} \linebreak
\noindent\hyperlink{ClassifyingSpaces}{Classifying spaces}\dotfill \pageref*{ClassifyingSpaces} \linebreak
\noindent\hyperlink{CechNerves}{Cech nerves}\dotfill \pageref*{CechNerves} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{ReferencesCompatibilityHomotopyPullback}{(Non-)Compatibility with homotopy pullbacks}\dotfill \pageref*{ReferencesCompatibilityHomotopyPullback} \linebreak


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

For $X_\bullet$ a [[simplicial object]] in [[Top]] -- a [[simplicial topological space]] -- its \emph{[[geometric realization]]} is a plain [[topological space]] ${|X_\bullet|} \in Top$ obtained by gluing all topological space $X_n$ together, as determined by the face and degeneracy maps.

The construction of ${|X_\bullet|}$ is a direct analog of the ordinary notion of [[geometric realization]] of a [[simplicial set]], but taking into account the [[topology]] on the spaces of $n$-simplices $X_n$.

The [[duality|dual]] concept is that of [[totalization]] of [[cosimplicial topological spaces]].

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

Let [[Top]] in the following denote either

\begin{itemize}%
\item the [[category]] of [[compactly generated space|compactly generated]] [[weakly Hausdorff space]]s, or


\item the category of [[k-space]]s.



\end{itemize}
Let $\Delta$ denote the [[simplex category]], and write $\Delta_{Top} \colon \Delta \to Top \colon  [n] \mapsto \Delta^n_{Top}$ for the standard [[cosimplicial object|cosimplicial]] [[topological space]] of topological [[simplices]].

Write equivalently

\begin{displaymath}
Top^{\Delta^{op}} \coloneqq  sTop \coloneqq  [\Delta^{op}, Top]
\end{displaymath}
for the category of [[simplicial topological spaces]].

\begin{defn}
\label{GeometricRealization}\hypertarget{GeometricRealization}{}
For $X_\bullet \colon  \Delta^{op} \to Top$ a [[simplicial topological space]], its \textbf{geometric realization} is the [[coend]]

\begin{displaymath}
{|X_\bullet|} \coloneqq  \int^{n \in \Delta} X_n \times \Delta^n_{Top}
\end{displaymath}
formed in [[Top]].

\end{defn}
This operation naturally extends to a [[functor]]

\begin{displaymath}
{|-|} \colon sTop \to Top
  \,.
\end{displaymath}
More explicitly, $\vert X_\bullet \vert$ is the [[topological space]] given by the [[quotient]]

\begin{displaymath}
{\vert X_\bullet \vert} = \coprod_{n} X_n \times \Delta^n_{Top} /\sim
\end{displaymath}
where the [[equivalence relation]] ``$\sim$'' identifies, for every morphism $[k] \to [l]$ in $\Delta$, the points $(x,f_* p) \in X_l \times \Delta^l_{Top}$ and $(f^* x,p) \in X_k \times \Delta^k_{Top}$.

This form of geometric realization of simplicial topological spaces goes back to (\hyperlink{Segal68}{Segal68}). An early reference that realizes this construction as a [[coend]] is (\hyperlink{MacLane}{MacLane}).

One also considers geometric realization after restricting to the subcategory $\Delta_+ \hookrightarrow \Delta$ of the [[simplex category]] on the strictly increasing maps (that is, the coface maps only---no codegeneracies).

\begin{defn}
\label{FatGeometricRealization}\hypertarget{FatGeometricRealization}{}
The corresponding [[coend]] in [[Top]] is called the \textbf{fat geometric realization}

\begin{displaymath}
{\Vert X_\bullet\Vert} \coloneqq   \int^{n \in \Delta_+} X_n \times \Delta^n_{Top}
  \,.
\end{displaymath}
\end{defn}
This is called \emph{fat} , because it does not [[quotient]] out the [[relation]]s induced by the degeneracy maps and hence is ``bigger'' than ordinary geometric realization.

Explicitly, this is the [[topological space]] given by the [[quotient]]

\begin{displaymath}
{\Vert X_\bullet \Vert} = \coprod_{n} X_n\times \Delta^n_{Top} /{\sim_+}
\end{displaymath}
where the [[equivalence relation]] ``$\sim_+$'' identifies $(x,f_* p) \in X_l\times \Delta^l_{Top}$ with $(f^* x,p) \in X_k\times \Delta^k_{Top}$ only when $[k] \to [l]$ is a coface map.

\begin{example}
\label{GeometricRealizationOfThePoint}\hypertarget{GeometricRealizationOfThePoint}{}
The geometric realization of the point --- the [[simplicial topological space]] that is in each degree the 1-point topological space --- is [[homeomorphic]] to the point, but the fat geometric realization of the point is an ``infinite dimensional topological ball'': the [[terminal object|terminal]] morphism

\begin{displaymath}
{\vert * \vert} \stackrel{\simeq_{iso}}{\longrightarrow} *
\end{displaymath}
is an [[isomorphism]], but the morphism

\begin{displaymath}
{\Vert * \Vert} \stackrel{\simeq_{h.e.}}{\longrightarrow} *
\end{displaymath}
is just a [[homotopy equivalence]].

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
The ordinary geometric realization can be described as the [[tensor product of functors]] ${|X_\bullet|}=\Delta^\bullet \otimes_{\Delta} X_\bullet$, and the fat geometric realization can likewise be described as ${\Vert X_\bullet\Vert} = i^* \Delta^\bullet \otimes_{\Delta_+} i^* X_\bullet$, where $i\colon \Delta_+ \hookrightarrow \Delta$ is the inclusion. By general facts about tensor product of functors (essentially, the associativity of composition of [[profunctors]]), it follows that we can also write ${\Vert X_\bullet\Vert} \cong \Delta^\bullet \otimes_{\Delta} \Lan_i i^* X_\bullet \cong {| Lan_i i^* X_\bullet |}$. In other words, the fat geometric realization of $X_\bullet$ is the ordinary geometric realization of a ``fattened up'' version of $X_\bullet$, which is obtained by forgetting the degeneracy maps of $X_\bullet$ and then ``freely throwing in new ones.''

\end{remark}
\hypertarget{reminder_on_nice_simplicial_topological_spaces}{}\subsection*{{Reminder on nice simplicial topological spaces}}\label{reminder_on_nice_simplicial_topological_spaces}

Simplicial topological spaces are in [[homotopy theory]] presentations for certain [[topological ∞-groupoid]]s . In this context what matters is not the operation of geometric realization itself, but its [[derived functor]]. This is obtained by evaluating ordinary geometric realization on ``sufficiently nice'' [[resolution]]s of simplicial topological spaces. These we discuss now.

Recall the following definitions and facts from [[nice simplicial topological space]].

\begin{defn}
\label{GoodAndProper}\hypertarget{GoodAndProper}{}
Let $X \colon  \Delta^{op} \to Top$ be a [[simplicial topological space]].

Such $X$ is called

\begin{itemize}%
\item \textbf{good} if all the degeneracy maps $X_{n-1} \hookrightarrow X_n$ are all [[closed cofibration]]s;


\item \textbf{proper} if the inclusion $s X_n \hookrightarrow X_n$ of the degenerate simplices is a [[closed cofibration]], where $s X_n = \bigcup_i s_i(X_{n-1})$.



\end{itemize}
Noticing that the union of degenerate simplices appearing here is a [[Reedy model structure|latching object]] and that closed cofibrations are cofibrations in the [[Strøm model structure]] on [[Top]], the last condition equivalently says:

\begin{itemize}%
\item $X_\bullet$ is \textbf{proper} if it is cofibrant in the [[Reedy model structure]] $[\Delta^{op}, Top_{Strom}]_{Reedy}$ on [[simplicial object]]s in [[Top]] with respect to the [[Strøm model structure]] on [[Top]].

\end{itemize}
\end{defn}
The notion of good simplicial topological space goes back to (\hyperlink{Segal73}{Segal73}), that of proper simplicial topological space to (\hyperlink{May}{May}).

\begin{prop}
\label{GoodImpliesProper}\hypertarget{GoodImpliesProper}{}
A good simplicial topological space is proper:

\begin{displaymath}
(X_\bullet \; good) \Rightarrow (X_\bullet\; proper)
  \,.
\end{displaymath}
\end{prop}
A proof appears as (\hyperlink{Lewis}{Lewis, corollary 2.4 (b)}). A generalization of this result to more general topological categories is (\hyperlink{RobertsStevenson}{RobertsStevenson, prop. 16}).

\hypertarget{bisimplicial_sets_and_good_resolutions}{}\subsubsection*{{Bisimplicial sets and good resolutions}}\label{bisimplicial_sets_and_good_resolutions}

We now discuss the [[resolution]] of any simplicial topological space by a good one. Write

\begin{displaymath}
({\vert- \vert} \dashv Sing) 
   \colon  
  Top
    \stackrel{\overset{{|-|}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}} 
  sSet
\end{displaymath}
for the ordinary [[geometric realization]]/[[singular simplicial complex]] [[adjunction]] (see [[homotopy hypothesis]]). Recall that the composite ${|Sing(-)|}\colon Top\to Top$ is a cofibrant replacement functor: for any space $X$, the space ${|Sing(X)|}$ is a CW complex and comes with a natural map ${|Sing X|} \to X$ (the [[unit of an adjunction|counit]] of the adjunction $({|-|} \dashv Sing)$) which is a [[weak homotopy equivalence]].

\begin{prop}
\label{GoodResolution}\hypertarget{GoodResolution}{}
Let $X_\bullet$ be a simplicial topological space. Then the simplicial topological space

\begin{displaymath}
{|Sing(X_\bullet)|} \in [\Delta^{op}, Top] \,,
\end{displaymath}
obtained by applying ${|Sing(-)|} \colon  Top \to Top$ degreewise, is good and hence proper. Moreover, we have a natural morphism

\begin{displaymath}
{|Sing X_\bullet|} \to X_\bullet
\end{displaymath}
which is degreewise a weak homotopy equivalence.

\end{prop}
\begin{proof}
Each space $|Sing X_n|$ is a [[CW-complex]], hence in particular a [[locally equi-connected space]]. By (\hyperlink{Lewis}{Lewis, p. 153}) inclusions of [[retract]]s of locally equi-connected spaces are [[closed cofibration]]s, and since degeneracy maps are retracts, this means that the degeneracy maps in $|Sing X_\bullet|$ are closed cofibrations.

The second sentence follows directly by the remarks above.

\end{proof}
Note that there is nothing special about ${|Sing(-)|}$ in the proof; any functorial CW replacement would do just as well (such as that obtained by the [[small object argument]]). However, ${|Sing(-)|}$ has the advantage that its geometric realization can be computed alternately in terms of diagonals of bisimplicial sets, as we now show.

If $S_{\bullet,\bullet} \colon  \Delta^{op} \times \Delta^{op} \to Set$ is a [[bisimplicial set]], we write $d S$ for its [[diagonal]], which is the composite

\begin{displaymath}
\Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\longrightarrow} Set.
\end{displaymath}
On the other hand, we can also consider a bisimplicial set as a simplicial object in $sSet$ and take its ``geometric realization'':

\begin{displaymath}
{|S_{\bullet,\bullet}|} \coloneqq \int^{n\in\Delta^{op}} S_{n,\bullet} \times \Delta^n_{sSet}
\end{displaymath}
where $\Delta^n_{sSet}$ denotes the $n$-simplex as a simplicial set, i.e. the representable functor $\Delta(-,[n])\colon \Delta^{op}\to Set$.

\begin{lemma}
\label{DiagonalIsRealization}\hypertarget{DiagonalIsRealization}{}
For a bisimplicial set $S_{\bullet,\bullet}$, there is a natural isomorphism of simplicial sets $d S \cong {|S|}$.

\end{lemma}
\begin{proof}
Both functors $d$ and ${|-|}$ are [[cocontinuous functor|cocontinuous]] functors between [[presheaf categories]] $Set^{\Delta^{op}\times\Delta^{op}} \to Set^{\Delta^{op}}$, so it suffices to verify that they agree on representables. The representable $(\Delta\times\Delta)(-,([n],[m]))$ is called a \emph{bisimplex} $\Delta^{n,m}$; its diagonal is

\begin{displaymath}
(d \Delta^{n,m})(-) = \Delta(-,n) \times \Delta(-,m) = \Delta^n_{sSet} \times \Delta^m_{sSet}.
\end{displaymath}
Now note that geometric realization of a bisimplicial set is left adjoint to the ``singular complex'' $Sing\colon sSet \to sSet^{\Delta^{op}}$ defined by

\begin{displaymath}
Sing(X_\bullet)(n,m) = \underline{Map}(\Delta^n_{sSet}, X_\bullet)_m = sSet(\Delta^n_{sSet} \times \Delta^m_{sSet}, X_\bullet) = sSet(d \Delta^{n,m}, X_\bullet)
\end{displaymath}
where $\underline{Map}(-,-)$ denotes the simplicial mapping space between two simplicial sets. But by the Yoneda lemma, $Sing(X_\bullet)(n,m) = sSet^{\Delta^{op}}(\Delta^{n,m},Sing(X_\bullet))$, so this shows that $d \Delta^{n,m}$ has the same universal property as ${|\Delta^{n,m}|}$; hence they are isomorphic.

\end{proof}
In particular, the lemma implies that for the two different ways of considering a bisimplicial set as a simplicial simplicial set (``vertically'' or ``horizontally''), the resulting ``geometric realizations'' as simplicial sets are isomorphic (since both are isomorphic to the diagonal, which is symmetrically defined).

Note that Lemma \ref{DiagonalIsRealization} can be interpreted as an isomorphism between two [[profunctors]] $\Delta &#8696; \Delta\times\Delta$, of which the first is [[representable functor|representable]] by the diagonal functor $\Delta \to \Delta\times\Delta$. It follows that if $S_{\bullet,\bullet}$ is a bisimplicial object in any [[cocomplete category]], we also have

\begin{displaymath}
d S_{\bullet,\bullet} \cong \int^{n\in\Delta} S_{n,\bullet}\,\times \,(\Delta^n_{sSet})_{\bullet}
\end{displaymath}
where the right-hand side is a ``realization'' functor from bisimplicial objects to simplicial objects in any cocomplete category. On the other hand, if $S$ is a bisimplicial \emph{space}, then we also have the \emph{levelwise} realization

\begin{displaymath}
\int^{n\in\Delta} S_{n,\bullet} \,\times\, \Delta^n_{Top}
\end{displaymath}
which will not, in general, agree with the diagonal and the abstract realization considered above. It does agree, however, after we pass to a further geometric realization as a single topological space.

\begin{prop}
\label{RealizationOfDiagonalIsRealizationOfRealization}\hypertarget{RealizationOfDiagonalIsRealizationOfRealization}{}
For $S_{\bullet,\bullet}$ any bisimplicial space, there is a [[homeomorphism]] between the geometric realizations of the following two simplicial spaces: (1) the diagonal $d S_{\bullet,\bullet}$ of $S$, and (2) the levelwise realization $|S_{\bullet,\bullet}|$ of $S$.

\end{prop}
\begin{proof}
Applying Lemma \ref{DiagonalIsRealization} as above, we have

\begin{displaymath}
d S_{\bullet,\bullet} \cong \int^{n\in \Delta} S_{n,\bullet} \times \Delta^n_{sSet}
\end{displaymath}
If we then take the realization of these simplicial spaces, we find

\begin{displaymath}
{| d S_{\bullet,\bullet} |}
\cong \left| \int^{n\in\Delta} S_{n,\bullet} \times \Delta^n_{sSet} \right|
\cong \int^{n\in\Delta} {|S_{\bullet,\bullet}|} \times \Delta^n_{Top}
\cong {\vert({\vert S_{\bullet,\bullet} \vert})\vert}
\end{displaymath}
using the fact that geometric realization of simplicial sets preserves colimits and products, and ${|\Delta^n_{sSet}|} = \Delta^n_{Top}$.

\end{proof}
This fact is attributed to Tornehave by Quillen on page 94 of his `Higher Algebraic K-theory I'.

Finally, for any simplicial space $X_\bullet$, we have a bisimplicial set $Sing(X_\bullet)_\bullet$. Applying the previous proposition to this bisimplicial set, regarded as a discrete bisimplicial space, we find a homeomorphism

\begin{displaymath}
{\vert({\vert Sing(X_\bullet) \vert})\vert}  \cong {| d Sing(X_\bullet)_\bullet |}
  \,.
\end{displaymath}
This also follows from results of (\hyperlink{Lewis}{Lewis}). Thus, as a resolution of $X_\bullet$, the levelwise realization of the levelwise singular complex ${\vert Sing(X_\bullet) \vert}$ has the pleasant property that its geometric realization, as a simplicial \emph{space}, can be calculated as the realization of a single simplicial \emph{set} (the diagonal of $Sing(X_\bullet)_\bullet$).

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

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

\begin{remark}
\label{}\hypertarget{}{}
We have the following degenerate case of geometric realization of simplicial topological spaces.

\begin{itemize}%
\item If $X_\bullet \colon  \Delta^{op} \to Set \hookrightarrow Top$ is a degreewise [[discrete space]], hence just a [[simplicial set]], the notion of geometric realization above coincides with the notion of [[geometric realization|geometric realization of simplicial sets]].


\item If $X_\bullet$ is simplicially constant on a topological space $X_0$, then its geometric realization is [[homeomorphic]] to that space:

\begin{displaymath}
{\vert X_\bullet \vert} \simeq X_0
  \,.
\end{displaymath}
(This is not true for the fat geometric realization, only the ordinary one. The fat geometric realization will be homotopy equivalent to $X$.)



\end{itemize}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Geometric realization of simplicial topological spaces has a [[right adjoint|right]] [[adjoint functor]] $\underline{Sing}$:

\begin{displaymath}
({\vert - \vert} \dashv \underline{Sing})  \colon 
   sTop
   \stackrel{\overset{{\vert - \vert}}{\longrightarrow}}{\underset{\underline{Sing}} {\longleftarrow}}
   Top
  \,,
\end{displaymath}
For $X \in$ [[Top]] a [[topological space]], we have by definition

\begin{displaymath}
\underline{Sing}(X) \colon  [n] \mapsto [\Delta^n_{Top}, X]
  \,,
\end{displaymath}
where on the right we have the [[internal hom]], or [[exponential law for spaces|exponential]], space from the standard topological $n$-simplex to $X$.

\end{remark}
\begin{prop}
\label{RealizationOfSingularComplexIsWHE}\hypertarget{RealizationOfSingularComplexIsWHE}{}
For every $X \in Top$ there is a [[weak homotopy equivalence]]

\begin{displaymath}
{|\underline{Sing}(X)|} \to X
  \,.
\end{displaymath}
\end{prop}
 The proof can be found in [[Neil Strickland]]`s answer to this \href{http://mathoverflow.net/q/171662/381}{mathoverflow question}. (An incorrect argument appears as (\hyperlink{Seymour}{Seymour, prop. 3.1}) where it is claimed that $\underline{Sing}(X)$ is proper. In fact, the first degeneracy map $X \to [\Delta^1_{Top}, X]$ is not in general a cofibration as explained in the linked question.)

Ordinary geometric realization has the following two disadvantages:

\begin{prop}
\label{RealizationDoesntPreserveMCofibrancyOrHEs}\hypertarget{RealizationDoesntPreserveMCofibrancyOrHEs}{}
\begin{itemize}%
\item If $X_\bullet$ has in each degree the [[homotopy type]] of a [[CW-complex]], its realization ${\vert X_\bullet \vert}$ in general need not.


\item If a morphism $f \colon  X_\bullet \to Y_\bullet$ is degreewise a [[homotopy equivalence]], its geometric realization ${\vert f \vert}$ need not be a homotopy equivalence.



\end{itemize}
\end{prop}
See (\hyperlink{Segal74}{Segal74, appendix A})

This is different for the fat geometric realization.

\begin{prop}
\label{NicePropertiesOfFatRealization}\hypertarget{NicePropertiesOfFatRealization}{}
\begin{itemize}%
\item If $X_\bullet$ is degreewise of the [[homotopy type]] of a [[CW-complex]] (i.e. is degreewise [[m-cofibrant space|m-cofibrant]]), then so is ${\Vert X_\bullet \Vert}$.


\item If $f \colon  X_\bullet \to Y_\bullet$ is degreewise a [[homotopy equivalence]], then also ${\Vert f \Vert}$ is a homotopy equivalence.



\end{itemize}
\end{prop}
This appears as (\hyperlink{Segal}{Segal74, prop. A.1}).

\hypertarget{relation_between_fat_and_ordinary_geometric_realization}{}\subsubsection*{{Relation between fat and ordinary geometric realization}}\label{relation_between_fat_and_ordinary_geometric_realization}

\begin{prop}
\label{FatRealizationOfGoodSimplicialSpaces}\hypertarget{FatRealizationOfGoodSimplicialSpaces}{}
If the simplicial topological space $X_\bullet$ is \hyperlink{GoodAndProper}{good} then the natural morphism from its \hyperlink{FatGeometricRealization}{fat geometric realization} to its \hyperlink{GeometricRealization}{ordinary geometric realization} is a [[homotopy equivalence]]

\begin{displaymath}
{\Vert X_\bullet \Vert} \stackrel{\simeq}{\longrightarrow} {|X_\bullet|}
  \,.
\end{displaymath}
\end{prop}
A direct proof of this (not using that good implies proper) appears as (\hyperlink{Segal74}{Segal74, prop. A.1 (iv)}) and a more detailed proof has been given by Tammo tom Dieck. A recent paper treating the special case where $X_\bullet$ is the nerve of a [[topological category]], and each $X_n$ is of the [[homotopy type]] of a [[CW-complex]], is in (\hyperlink{Wang17}{Wang 2017}).

\hypertarget{compatibility_with_limits}{}\subsubsection*{{Compatibility with limits}}\label{compatibility_with_limits}

We discuss how geometric realization interacts with [[limit]]s of simplicial topological spaces.

\hypertarget{ordinary_geometric_realization}{}\paragraph*{{Ordinary geometric realization}}\label{ordinary_geometric_realization}

\begin{prop}
\label{RealizationPreservesPullbacks}\hypertarget{RealizationPreservesPullbacks}{}
Geometric realization preserves [[pullback]]s: for $X_\bullet \to Y_\bullet \leftarrow Z_\bullet$ a [[diagram]] in $Top^{\Delta^{op}}$ there are [[natural transformation|natural]] [[homeomorphism]]s

\begin{displaymath}
{|X_\bullet|} \,\times_{|Y_\bullet|}\, {|Z_\bullet|}
  \simeq
  {|X_\bullet \,\times_{Y_\bullet}\, Z_\bullet|}
  \,.
\end{displaymath}
\end{prop}
This appears for instance as (\hyperlink{May}{May, corollary 11.6}). See also the proof that geometric realization of simplicial sets preserves pullbacks, at [[geometric realization]].

It is essential here that we are working in a category $Top$ such as compactly generated spaces or k-spaces: in the category of \emph{all} topological spaces this would not be true. It works in these cases because product and/or quotient topologies in these categories are slightly different from in the category of all topological spaces.

\hypertarget{fat_geometric_realization}{}\paragraph*{{Fat geometric realization}}\label{fat_geometric_realization}

The operation of \hyperlink{FatGeometricRealization}{fat geometric realization} does not preserve [[fiber product]]s on the nose, in general, but it does preserve all [[finite limit]]s up to [[homotopy]].

Write ${\Vert * \Vert}$ for the fat geometric realization of the point. Notice that due to the identification of $sTop$ with its [[overcategory]] over the point (the simplicial topological space constant on the point), $sTop\simeq sTop/*$, we may regard fat geometric gealization as a functor with values in the [[overcategory]] $Top/{\Vert*\Vert}$ over the \hyperlink{GeometricRealizationOfThePoint}{fat geometric realization of the point}.

\begin{prop}
\label{FatRealizationPreservesConnectedFiniteLimits}\hypertarget{FatRealizationPreservesConnectedFiniteLimits}{}
The functor

\begin{displaymath}
{\Vert - \Vert} \colon  Top^{\Delta^{op}} \to Top/{\Vert*\Vert}
\end{displaymath}
preserves all [[finite limit]]s.

\end{prop}
See (\hyperlink{GepnerHenriques07}{Gepner-Henriques 07, Remark 2.23}).

\hypertarget{RelationToHomotopyColimit}{}\subsubsection*{{Relation to the homotopy colimit}}\label{RelationToHomotopyColimit}

Recall that a simplicial topological space is proper if it is Reedy cofibrant relative to the [[Strøm model structure]] on [[Top]], in which the weak equivalences are the honest [[homotopy equivalences]]. Nevertheless, in certain cases geometric realisation computes the [[homotopy colimit]] of the [[diagram]] $X_\bullet \colon  \Delta^{op} \to Top$ given by the simplicial space, with respect to the standard [[Quillen model structure on topological spaces]] in which the weak equivalences are the [[weak homotopy equivalences]].

\begin{lemma}
\label{RealizPresWHE}\hypertarget{RealizPresWHE}{}
If $X_\bullet \to Y_\bullet$ is an objectwise weak homotopy equivalence between proper simplicial spaces, then the induced map ${|X_\bullet|} \to {|Y_\bullet|}$ is a weak homotopy equivalence.

\end{lemma}
\begin{proof}
The following proof is essentially from (\hyperlink{May_Einf}{May74, A.4}); see also (\hyperlink{Dugger}{Dugger, prop. 17.4, example 18.2}). It relies on two facts relating [[Hurewicz cofibration]]s to [[weak homotopy equivalence]]s:

\begin{enumerate}%
\item [[pushout|Pushouts]] along [[Hurewicz cofibration]]s preserve [[weak homotopy equivalence]]s, and


\item [[colimit|Colimits]] of sequences of Hurewicz cofibrations preserve weak homotopy equivalences.



\end{enumerate}
Let $i_n \colon \Delta_{\le n} \hookrightarrow \Delta$ denote the inclusion of the objects $\le n$, and write ${|X_\bullet|}_n = i_n^* \Delta \otimes_{\Delta_{\le n}} i_n^* X_\bullet$. Writing $L_n X$ for the $n$th [[latching object]] (the subspace of degeneracies in $X_n$), we have pushouts

\begin{displaymath}
\itexarray{ (L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) & \to &  {|X_\bullet|}_{n-1}\\
  \downarrow & & \downarrow\\
  X_n \times \Delta^n & \to &  {|X_\bullet|}_n }
\end{displaymath}
Since $X$ is proper, $L_n X \to X_n$ is a cofibration, and of course $\partial\Delta^n \to \Delta^n$ is a cofibration. Thus, by the [[pushout-product axiom]] for the [[Strøm model structure]], the left-hand vertical map is a cofibration; hence so is the right-hand vertical map.

Now ${|X_\bullet|}$ is the [[colimit]] of the sequence of cofibrations

\begin{displaymath}
{|X_\bullet|}_0 \to {|X_\bullet|}_1 \to {|X_\bullet|}_2 \to \cdots
\end{displaymath}
and likewise for ${|Y_\bullet|}$. In other words, the geometric realization is [[filtration|filtered]] by simplicial degree. Thus, by point (2) above, it suffices to show that each map ${|X_\bullet|}_n \to {|Y_\bullet|}_n$ is a weak homotopy equivalence.

Since ${|X_\bullet|}_0 = X_0$, this is true for $n=0$. Moreover, by the above pushout square, ${|X_\bullet|}_n$ is a pushout of ${|X_\bullet|}_{n-1}$ along a cofibration. Thus, by point (1) above, since $X_n \times \Delta^n \to Y_n \times \Delta^n$ is certainly a weak homotopy equivalence, it will suffice for an induction step to prove that

\begin{displaymath}
(L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) \to
(L_n Y \times \Delta^n) \sqcup_{L_n Y \times \partial\Delta^n} (Y_n \times \partial\Delta^n)
\end{displaymath}
is a weak homotopy equivalence. However, by definition we have a pushout

\begin{displaymath}
\itexarray{
  L_n X \times \partial\Delta^n & \to & X_n \times \partial\Delta^n \\
  \downarrow & & \downarrow\\
  L_n X \times \Delta^n & \to &
  (L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) }
\end{displaymath}
This is also a pushout along a Hurewicz cofibration, and cartesian product preserves weak homotopy equivalences, so it will suffice to show that $L_n X \to L_n Y$ is a weak homotopy equivalence.

Recall that $L_n X$ can be written as $\colim^{[n]\to [k]} X_{[k]}$, where the colimit is over all codegeneracy maps $[n]\to [k]$ in $\Delta$ \emph{except} the identity $[n] \to [n]$. For $0\le m \le n$, write $L^m_n X$ for the corresponding colimit over all codegeneracies which factor through $\sigma_{p}\colon [n]\to [n-1]$ for some $0\le p\le m$. Then $L^0_n X = X_{n-1}$ and $L^n_n X = L_n X$, and for $0\lt m \le n$ we have a pushout square

\begin{displaymath}
\itexarray{ L^{m-1}_{n-1} X & \to & L^{m-1}_n X \\
  \downarrow & & \downarrow \\
  X_{n-1} & \to & L^m_n X. }
\end{displaymath}
We claim that $L^{m-1}_n X \to L^m_n X$ is a cofibration for all $0\le m \le n$, and we prove it by induction on $n$. For $n=0$ it is obvious. If it holds for $(n-1)$ (and all $m$ with $0\le m \le n-1$), then by composition and properness of $X$, each map $L^{m-1}_{n-1} X \to X_{n-1}$ is a cofibration. Hence, by the above pushout square, so is $L^{m-1}_n X \to L^m_n X$. This proves the claim.

Now, using the above pushout square again and point (1) above, we can prove by induction on $n$, and for fixed $n$, by induction on $m$, that each map $L^m_n X \to L^m_n Y$ is a weak homotopy equivalence. In particular, taking $m=n$, we find that $L_n X \to L_n Y$ is a weak homotopy equivalence, as desired.

\end{proof}
Recall that one way to compute the homotopy colimit of a diagram $X\colon D^{op}\to Top$, with respect to the standard (Quillen) model structure, is as the tensor product

\begin{displaymath}
hocolim X \coloneqq N(D/ -) \otimes_D Q X,
\end{displaymath}
where $(D/ -)\colon D \to Cat$ sends each object of $D$ to its [[overcategory]], $N$ denotes the [[nerve]] of a small category, and $Q$ denotes a functorial cofibrant replacement in the [[Quillen model structure on topological spaces]] (e.g. [[CW complex|CW]] replacement via singular [[nerve and realization]]). When $D=\Delta$, there is a canonically defined map $N(\Delta / -) \to \Delta$ (where the second $\Delta$ denotes the canonical cosimplicial simplicial set) called the [[Bousfield-Kan map]]. This map induces, for each simplicial space $X$, a map

\begin{displaymath}
N(D/ -) \otimes_D X_\bullet \to \Delta \otimes_{\Delta} X_\bullet = {|X_\bullet|}
\end{displaymath}
which is also called the \textbf{Bousfield-Kan map}.

Since the Str\o{}m model structure is a [[simplicial model category]], standard arguments involving Reedy model structures imply that the Bousfield-Kan map is a Str\o{}m weak equivalence (i.e. a homotopy equivalence) whenever $X$ is Str\o{}m Reedy cofibrant (i.e. proper). Thus we have:

\begin{prop}
\label{RealizationOfGoodSimplicialSpacesIsHomotopyColimit}\hypertarget{RealizationOfGoodSimplicialSpacesIsHomotopyColimit}{}
Let $X_\bullet$ be a proper simplicial space. Then the composite

\begin{displaymath}
\hocolim X_\bullet = N(D/ -) \otimes_D Q X_\bullet \xrightarrow{Bousfield-Kan} {|Q X_\bullet|} \longrightarrow {|X_\bullet|}
\end{displaymath}
is a weak homotopy equivalence.

\end{prop}
\begin{proof}
Since the definition of $\hocolim$ doesn't depend on the choice of an objectwise-cofibrant replacement of $X$, we may as well take $Q$ to be, instead of the composite with a functorial cofibrant replacement in $Top_{Quillen}$, rather a cofibrant replacement in the Reedy model structure on $Top^{\Delta^{op}}$ with respect to the Quillen model structure on $Top$. (Any Reedy cofibrant diagram is in particular objectwise cofibrant.) Then $Q X_\bullet$ is Quillen-Reedy cofibrant, hence also proper, and so the Bousfield-Kan map is a homotopy equivalence. On the other hand, $Q X_\bullet \to X_\bullet$ is a levelwise weak homotopy equivalence, while $Q X_\bullet$ and $X_\bullet$ are proper, so by Lemma \ref{RealizPresWHE} its realization is a weak homotopy equivalence.

\end{proof}
By naturality, the above composite is also equal to the composite

\begin{displaymath}
\hocolim X_\bullet = N(D/ -) \otimes_D Q X_\bullet \to N(D/ -) \otimes_D X_\bullet \xrightarrow{Bousfield-Kan} {|X_\bullet|};
\end{displaymath}
hence this composite is also a weak homotopy equivalence.

\hypertarget{examples_and_applications}{}\subsection*{{Examples and Applications}}\label{examples_and_applications}

\hypertarget{TopologicalPrincipalInfinityBundle}{}\subsubsection*{{Topological principal $\infty$-bundles}}\label{TopologicalPrincipalInfinityBundle}

We discuss aspects of [[principal ∞-bundle]]s equipped with topological [[cohesive (∞,1)-topos|cohesion]] and their geometric realization to [[principal bundle]]s in [[Top]].

For $X,Y \in sTop$, write

\begin{displaymath}
sTop(X,Y) \coloneqq \int_{[k] \in \Delta} [X_k, Y_k] \;\; \in Top
\end{displaymath}
for the [[Top]]-[[hom-object]], where in the integrand of the [[end]] $[-,-] \colon  Top^{op} \times Top^{op} \to Top$ is the [[internal hom]] of topological spaces.

\begin{defn}
\label{GloballyKanSimplicialTopologicalSpace}\hypertarget{GloballyKanSimplicialTopologicalSpace}{}
We say a morphism $f \colon  X \to Y$ of [[simplicial topological space]]s is a \textbf{global Kan fibration} if for all $n \in \mathbb{N}$ and $0 \leq i \leq n$ the canonical morphism

\begin{displaymath}
X_n \to Y_n \;\times_{sTop(\Lambda[n]_i, Y)}\; sTop(\Lambda[n]_i, X)
\end{displaymath}
in [[Top]] has a [[section]], where $\Lambda[n]_i \in$ [[sSet]] $\hookrightarrow Top^{\Delta^{op}}$ is the $i$th $n$-[[horn]] regarded as a [[discrete space|discrete]] [[simplicial topological space]].

We say a [[simplicial topological space]] $X_\bullet \in Top^{\Delta^{op}}$ is \textbf{(global) Kan simplicial space} if the unique morphism $X_\bullet \to *$ is a global Kan fibration, hence if for all $n \in \mathbb{N}$ and all $0 \leq i \leq n$ the canonical [[continuous function]]

\begin{displaymath}
X_n \to sTop(\Lambda[n]_i, X)
\end{displaymath}
into the [[topological space]] of $i$th $n$-[[horn]]s admits a [[section]] (in [[Top]], hence a global, continuous section).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This global notion of topological Kan fibration is considered in (\hyperlink{BrownSzczarba}{BrownSzczarba, def. 2.1, def. 6.1}). In fact there a stronger condition is imposed: a [[Kan complex]] in [[Set]] automatically has the lifting property not only against all full [[horn]] inclusions but also against sub-horns; and in (\hyperlink{BrownSzczarba}{BrownSzczarba}) all these fillers are required to be given by global sections. This ensures that with $X$ globally Kan also the [[internal hom]] $[Y,X] \in sTop$ is globally Kan, for any simplicial topological space $Y$. This is more than we need and want to impose here. For our purposes it is sufficient to observe that if $f$ is globally Kan in the sense of (\hyperlink{BrownSzczarba}{BrownSzczarba, def. 6.1}), then it is so also in the \hyperlink{GloballyKanSimplicialTopologicalSpace}{above sense}.

\end{remark}
Recall from the discussion at [[universal principal ∞-bundle]] that for $G$ a [[simplicial topological group]] the [[universal simplicial principal bundle]] $\mathbf{E}G \to \mathbf{B}G$ is presented by the morphism of [[simplicial topological space]]s traditionally denoted $W G \to \bar W G$.

\begin{prop}
\label{SimplicialTopologicalUniversalBundle}\hypertarget{SimplicialTopologicalUniversalBundle}{}
Let $G$ be a [[simplicial topological group]]. Then

\begin{enumerate}%
\item $G$ is a globally Kan simplicial topological space;


\item $\bar W G$ is a globally Kan simplicial topological space;


\item $W G \to \bar W G$ is a global Kan fibration.



\end{enumerate}
\end{prop}
\begin{proof}
The first statement appears as (\hyperlink{BrownSzczarba}{BrownSzczarba, theorem 3.8}), the second is noted in (\hyperlink{RobertsStevenson}{RobertsStevenson}), the third appears as (\hyperlink{BrownSzczarba}{BrownSzczarba, lemma 6.7}).

\end{proof}
\begin{prop}
\label{BarWGIsGoodIfGIsWellSectioned}\hypertarget{BarWGIsGoodIfGIsWellSectioned}{}
If $G$ is a [[well-pointed simplicial topological group]] then both $W G$ and $\bar W G$ are \hyperlink{GoodAndProper}{good simplicial topological space}.

\end{prop}
For $\bar W G$ this is (\hyperlink{RobertsStevenson}{RobertsStevenson, prop. 19}). For $W G$ this follows with (\hyperlink{RobertsStevenson}{RobertsStevenson, lemma 10, lemma 11}) which says that $W G = Dec_0 \bar W G$ and the observations in the proof of (\hyperlink{RobertsStevenson}{RobertsStevenson, prop. 16}) that $Dec_0 X$ is good if $X$ is.

\begin{prop}
\label{RealizationSimplicialTopologicalUniversalBundle}\hypertarget{RealizationSimplicialTopologicalUniversalBundle}{}
For $G$ a [[well-pointed simplicial topological group]], the geometric realization of the [[universal simplicial principal bundle]] $W G \to \bar W G$

\begin{displaymath}
{\vert W G \vert} \to {\vert \bar W G \vert}
\end{displaymath}
is a fibration [[resolution]] in $Top_{Quillen}$ of the point inclusion $* \to B{|G|}$ into the [[classifying space]] of the geometric realization of $G$.

\end{prop}
This is (\hyperlink{RobertsStevenson}{RobertsStevenson, prop. 14}).

\begin{prop}
\label{SimplicialTopolgicalBundleIsGood}\hypertarget{SimplicialTopolgicalBundleIsGood}{}
Let $X_\bullet$ be a \hyperlink{GoodAndProper}{good} [[simplicial topological space]] and $G$ a [[well-pointed simplicial topological group]]. Then for every morphism

\begin{displaymath}
\tau \colon  X \to \bar W G
\end{displaymath}
the corresponding topological [[simplicial principal bundle]] $P$ over $X$ is itself a good simplicial topological space.

\end{prop}
\begin{proof}
The bundle is the [[pullback]] $P = X \times_{\bar W G} W G$ in $sTop$

\begin{displaymath}
\itexarray{
     P &\to& \bar W G
     \\
     \downarrow && \downarrow
     \\
     X &\stackrel{\tau}{\to}& \bar W G
  }
  \,.
\end{displaymath}
By assumption on $X$ and $G$ and using prop. \ref{BarWGIsGoodIfGIsWellSectioned} we have that $X$, $\bar W G$ and $W G$ are all good simplicial spaces.

This means that the degeneracy maps of $P_\bullet$ are induced degreewise by morphisms between pullbacks in [[Top]] that are degreewise [[closed cofibration]]s, where one of the morphisms in each pullback is a fibration. By the properties discussed at [[closed cofibration]], this implies that also these degeneracy maps of $P_\bullet$ are closed cofibrations.

\end{proof}
Let for the following $Top_s \hookrightarrow Top$ be any [[small category|small]] [[full subcategory]].

\begin{defn}
\label{SimpTopAsSimpPresheaves}\hypertarget{SimpTopAsSimpPresheaves}{}
Under the degreewise [[Yoneda embedding]] $sTop_s \hookrightarrow [Top_s^{op}, sSet]$ simplicial topological spaces embed into the category of [[simplicial presheaves]] on $Top_s$.

We equip this with the projective [[model structure on simplicial presheaves]] $[Top_s^{op}, sSet]_{proj}$, and we speak of [[homotopy limit]]s in $sTop$ under this embedding.

\end{defn}
\begin{prop}
\label{GlobalKanFibImpliesProjectiveFib}\hypertarget{GlobalKanFibImpliesProjectiveFib}{}
Under this embedding a \hyperlink{GloballyKanSimplicialTopologicalSpace}{global Kan fibration} $f \colon  X \to Y$ in $sTop_s$ maps to a fibration in $[Top_s^{op}, sSet]_{proj}$.

\end{prop}
\begin{proof}
By definition, a morphism $f \colon  X \to Y$ in $[Top_s^{op}, sSet]_{proj}$ is a fibration if for all $U \in Top$ and all $n \in \mathbb{N}$ and $0 \leq i \leq n$ diagrams of the form

\begin{displaymath}
\itexarray{
    \Lambda[n]_i \cdot U &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    \Delta[n] \cdot U &\to& Y
  }
\end{displaymath}
have a lift. This is equivalent to saying that the [[function]]

\begin{displaymath}
Hom(\Delta[n]\cdot U, X)
   \to 
  Hom(\Delta[n]\cdot U,Y) 
    \times_{Hom(\Lambda[n]_i \cdot U, Y)}
  Hom(\Lambda[n]_i \cdot U, X)
\end{displaymath}
is surjective. Notice that we have

\begin{displaymath}
\begin{aligned}
    Hom_{[Top_s^{op}, sSet]}(\Delta[n]\cdot U, X)
    & = 
    Hom_{sTop}(\Delta[n]\cdot U, X)
    \\
     & = \int_{[k] \in \Delta} Hom_{Top}( \Delta[n]_k \times U, X_k)
   \\
    & = \int_{[k] \in \Delta} Hom_{Top}(U, [\Delta[n]_k, X_k])
   \\
    & = Hom_{Top}(U, \int_{[k] \in \Delta} [\Delta[n]_k, X_k])
   \\
    & = Hom_{Top}(U, sTop(\Delta[n], X))
   \\
    & = Hom_{Top}(U, X_n)
  \end{aligned}
\end{displaymath}
and analogously for the other factors in the above morphism. Therefore the lifting problem equivalently says that the function

\begin{displaymath}
Hom_{Top}(U, \; X_n \to Y_n \times_{sTop(\Lambda[n]_i, Y)} sTop(\Lambda[n]_i,X) \;)
\end{displaymath}
is surjective. But by the assumption that $f \colon  X \to Y$ is a global Kan fibration of simplicial topological spaces, def. \ref{GloballyKanSimplicialTopologicalSpace}, we have a section $\sigma \colon  Y_n \times_{sTop(\Lambda[n]_i), Y} sTop(\Lambda[n]_i,X) \to X_n$. Therefore $Hom_{Top}(U, \sigma)$ is a section of our function.

\end{proof}
\begin{prop}
\label{HocolimPreservesHomotopyFibers}\hypertarget{HocolimPreservesHomotopyFibers}{}
The [[homotopy colimit]] operation

\begin{displaymath}
sTop \hookrightarrow [Top_s^{op}, sSet]_{proj} \stackrel{hocolim}{\to} Top_{Quillen}
\end{displaymath}
preserves [[homotopy fiber]]s of morphisms $\tau \colon  X \to \bar W G$ with $X$ good and globally Kan and $G$ well-pointed.

\end{prop}
\begin{proof}
By prop. \ref{SimplicialTopologicalUniversalBundle} and prop. \ref{GlobalKanFibImpliesProjectiveFib} we have that $W G \to \bar W G$ is a fibration [[resolution]] of the point inclusion $* \to \bar W G$ in $[Top^{op}, sSet]_{proj}$. By the general discussion at [[homotopy limit]] this means that the [[homotopy fiber]] of a morphism $\tau \colon  X \to \bar W G$ is computed as the ordinary [[pullback]] $P$ in

\begin{displaymath}
\itexarray{
    P &\to& W G
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\tau}{\to}& \bar W G
  }
\end{displaymath}
(since all objects $X$, $\bar W G$ and $W G$ are fibrant and at least one of the two morphisms in the pullback diagram is a fibration) and hence

\begin{displaymath}
hofib(\tau) \simeq P
  \,.
\end{displaymath}
By prop. \ref{SimplicialTopologicalUniversalBundle} and prop. \ref{SimplicialTopolgicalBundleIsGood} it follows that all objects here are good simplicial topological spaces. Therefore by prop. \ref{RealizationOfGoodSimplicialSpacesIsHomotopyColimit} we have

\begin{displaymath}
hocolim P_\bullet \simeq {|P_\bullet|}
\end{displaymath}
in [[Ho(Top)]]. By prop. \ref{RealizationPreservesPullbacks} we have that

\begin{displaymath}
\cdots = {|X_\bullet|} \times_{|\bar W G|} {|W G|}
  \,.
\end{displaymath}
But prop. \ref{RealizationSimplicialTopologicalUniversalBundle} says that this is again the presentation of a homotopy pullback/homotopy fiber by an ordinary pullback

\begin{displaymath}
\itexarray{
    {|P|} &\to& {|W G|}
    \\
    \downarrow && \downarrow
    \\
    {|X|} &\stackrel{\tau}{\to}& {|\bar W G|}
  }
  \,,
\end{displaymath}
because $|W G| \to |\bar W G|$ is again a fibration resolution of the point inclusion. Therefore

\begin{displaymath}
hocolim P_\bullet \simeq hofib( {|\tau|} )
  \,.
\end{displaymath}
Finally by prop. \ref{RealizationOfGoodSimplicialSpacesIsHomotopyColimit} and using the assumption that $X$ and $\bar W G$ are both good, this is

\begin{displaymath}
\cdots \simeq hofib (hocolim \tau)
  \,.
\end{displaymath}
In total we have shown

\begin{displaymath}
hocolim (hofib \tau) \simeq hofib (hocolim \tau)
  \,.
\end{displaymath}
\end{proof}
\hypertarget{ClassifyingSpaces}{}\subsubsection*{{Classifying spaces}}\label{ClassifyingSpaces}

For $G$ a [[topological group]], write $\mathbf{B}G$ for its [[delooping]] [[topological groupoid]]: the topological groupoid with a single object and $Mor_{\mathbf{B}G}(*,*) \coloneqq  G$, with composition given by the product on $G$.

The [[nerve]] $N \mathbf{B}G$ of this topological groupoid is naturally a [[simplicial topological space]], with

\begin{displaymath}
N \mathbf{B}G \colon  [n] \mapsto G^{\times_n}
  \,.
\end{displaymath}
\begin{prop}
\label{ClassifyingSpaceByGeometricRealization}\hypertarget{ClassifyingSpaceByGeometricRealization}{}
The geometric realization of $N \mathbf{B}G$ is a model for the [[classifying space]] $B G$ of $G$-[[principal bundle]]s

\begin{displaymath}
{|N \mathbf{B}G|} \simeq B G
  \,.
\end{displaymath}
\end{prop}
An early reference for this classical fact is (\hyperlink{Segal68}{Segal68}).

\begin{cor}
\label{OmegaBGisaWHE}\hypertarget{OmegaBGisaWHE}{}
Let $G$ be a [[well-pointed simplicial topological group|well-pointed]] [[topological group]], $B G$ its [[classifying space]] and $\Omega B G$ the [[loop space]] of the classifying space.

There is a [[weak homotopy equivalence]]

\begin{displaymath}
\Omega B G \stackrel{\simeq}{\to} G
  \,.
\end{displaymath}
\end{cor}
\begin{proof}
By prop. \ref{ClassifyingSpaceByGeometricRealization} we have that $B G \simeq |\mathbf{B}G|$ (we suppress the [[nerve]] operation notationally, which injects [[groupoid]]s into [[∞-groupoid]]s).

By standard facts about the $\bar W$-functor (see [[simplicial principal bundle]]) we have a [[pullback]] square of [[simplicial topological space]]s

\begin{displaymath}
\itexarray{
    G &\to& W G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \bar W G
  }
\end{displaymath}
exhibiting the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    G &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& \mathbf{B}G
  }
  \,.
\end{displaymath}
Under geoemtric realization this maps to

\begin{displaymath}
\itexarray{
    G &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& B G
  }
\end{displaymath}
in [[Top]]. By prop \ref{HocolimPreservesHomotopyFibers} this is still a homotopy pullback, and hence exhibits $G$ as the [[loop space]] of $B G$.

\end{proof}
\hypertarget{CechNerves}{}\subsubsection*{{Cech nerves}}\label{CechNerves}

\begin{prop}
\label{RealizationOfCechNerveIsWHE}\hypertarget{RealizationOfCechNerveIsWHE}{}
Let $X,Y$ be [[topological space]]s and $\pi \colon  Y \to X$ a [[continuous function]] that admits local [[section]]s. Write $C(\pi) \in sTop$ for the [[Cech nerve]].

Then the canonical map

\begin{displaymath}
{|C(\pi)|} \to X
\end{displaymath}
from the geometric realization of $X$ back to $X$ is a [[weak homotopy equivalence]]. If $X$ is a [[paracompact topological space]] then it is even a [[homotopy equivalence]].

\end{prop}
For paracompact $X$ this goes back to (\hyperlink{Segal68}{Segal68}). The general case is discussed in (\hyperlink{DuggerIsaksen}{DuggerIsaksen}). A generalization to parameterized spaces is in (\hyperlink{RobertsStevenson}{RobertsStevenson, lemma 22}).

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

\begin{itemize}%
\item [[simplicial topological space]], [[nice simplicial topological space]]


\item [[simplicial topological group]]


\item [[geometric realization]]

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

\end{itemize}


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

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

The first occurence of the definition of geometric realization of simplicial topological spaces seems to be

\begin{itemize}%
\item [[Graeme Segal]], \emph{Classifying spaces and spectral sequences} Publications Math\'e{}matiques de l'IH\'E{}S, 34 (1968), p. 105-112 (\href{http://www.numdam.org/item?id=PMIHES_1968__34__105_0}{numdam})

\end{itemize}
but the construction was implicit in earlier discussion of [[classifying space]]s. The observation that this is a [[coend]] was noted in

\begin{itemize}%
\item [[Saunders MacLane]], \emph{The Milgram bar construction as a tensor product of functors} SLNM Vol. 168 (1970)

\end{itemize}
The definition of \emph{good} simplicial topological spaces goes back to

\begin{itemize}%
\item [[Graeme Segal]], \emph{Configuration-Spaces and Iterated Loop-Spaces} , Inventiones math. 21,213-221 (1973)

\end{itemize}
An original reference on geometric realization of simplicial topological spaces is is appendix A of

\begin{itemize}%
\item [[Graeme Segal]], \emph{[[SegalCategoriesAndCohomologyTheories.pdf:file]]} Topology, 13:293--312, 1974

\end{itemize}
A proof that ordinary and fat geometric realisation give homotopic spaces, for the special case of the nerve of a topological category is in

\begin{itemize}%
\item Yi-Sheng Wang, \emph{Fat realization and Segal's classifying space}, arXiv:\href{https://arxiv.org/abs/1710.03796}{1710.03796}

\end{itemize}
A standard textbook reference is chapter 11 of

\begin{itemize}%
\item [[Peter May]], \emph{The geometry of iterated loop spaces} (\href{}{pdf})

\end{itemize}
A proof that good simplicial spaces are proper is implicit in the proof of lemma A.5 in (\hyperlink{Segal74}{Segal74}). Explicitly it appears in

\begin{itemize}%
\item L. Gaunce Lewis Jr., \emph{When is the natural map $X\to \Omega \Sigma X$ a cofibration?} , Trans. Amer. Math. Soc. \textbf{273} (1982) no. 1, 147--155 (\href{.org/pss/1999197}{JSTOR})

\end{itemize}
A generalization of the statement that good implies proper to other [[topological concrete categories]] and a discussion of the geometric realization of $W G \to \bar W G$ for $G$ a [[simplicial topological group]] is in

\begin{itemize}%
\item [[David Roberts]], [[Danny Stevenson]], \emph{Simplicial principal bundle in parameterized spaces} (\href{http://arxiv.org/abs/1203.2460}{arXiv:1203.2460})


\item [[Danny Stevenson]], \emph{Classifying theory for simplicial parametrized groups} (\href{http://arxiv.org/abs/1203.2461}{arXiv:1203.2461})



\end{itemize}
Comments on the relation between properness and cofibrancy in the [[Reedy model structure]] on $[\Delta^{op}, Set]$ are made in

\begin{itemize}%
\item [[Paul Goerss]], [[Kristen Schemmerhorn]], \emph{Model Categories and Simplicial Methods} (\href{http://www.math.northwestern.edu/~pgoerss/papers/ucnotes.pdf}{pdf}).

\end{itemize}
The relation between (fat) geometric realization and [[homotopy colimit]]s is considered as prop. 17.5 and example 18.2 of

\begin{itemize}%
\item [[Daniel Dugger]], \emph{A primer on homotopy colimits} (\href{http://pages.uoregon.edu/ddugger/hocolim.pdf}{pdf})

\end{itemize}
The proof that geometric realization of proper simplicial spaces preserves weak equivalences is from

\begin{itemize}%
\item [[Peter May]], \emph{$E_\infty$-spaces, group completions, and permutative categories}. London Math. Soc. Lecture Notes No. 11, 1974, 61-93.

\end{itemize}
A definition of the Bousfield-Kan map, and the Reedy model category theory necessary to show that it is a weak equivalence, can be found in

\begin{itemize}%
\item Hirschhorn, \emph{Model categories and their localizations}, AMS Mathematical Surveys and Monographs No. 99, 2003

\end{itemize}
The (fat) geometric realization of ([[nerves]] of) [[topological groupoid]]s is discussed in section 2.3 of

\begin{itemize}%
\item [[David Gepner]], [[André Henriques]], \emph{Homotopy theory of orbispaces} (\href{https://arxiv.org/abs/math/0701916}{arXiv:math/0701916})

\end{itemize}
See also

\begin{itemize}%
\item [[David Carchedi]], section 3.2 of \emph{On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces} (\href{http://arxiv.org/abs/1504.02394}{arXiv:1504.02394})

\end{itemize}
Globally Kan simplicial spaces are considered in

\begin{itemize}%
\item E. H. Brown and R. H. Szczarba, \emph{Continuous cohomology and real homotopy type} , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 (\href{http://www.ams.org/journals/tran/1989-311-01/S0002-9947-1989-0929667-6/S0002-9947-1989-0929667-6.pdf}{pdf})

\end{itemize}
The right adjoint to geometric realization of simplicial topological spaces is discussed in

\begin{itemize}%
\item R. M. Seymour, \emph{Kan fibrations in the category of simplicial spaces} Fund. Math., 106(2):141-152, 1980.

\end{itemize}
Geometric realization of general [[Cech nerve]]s is discussed in

\begin{itemize}%
\item [[Dan Dugger]], D. C. Isaksen, \emph{Topological hypercovers and $\mathbb{A}^1$- realizations, Math. Z. 246 (2004) no. 4}

\end{itemize}
\hypertarget{ReferencesCompatibilityHomotopyPullback}{}\subsubsection*{{(Non-)Compatibility with homotopy pullbacks}}\label{ReferencesCompatibilityHomotopyPullback}

Discussion of sufficient conditions for geometric realization to be compatible with [[homotopy pullbacks]]:

\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 realization of a simplicial space]] [[!redirects geometric realization of a simplicial topological space]] [[!redirects geometric realization of simplicial spaces]] [[!redirects fat geometric realization of simplicial spaces]] [[!redirects fat geometric realization of simplicial topological spaces]] [[!redirects fat geometric realization of a simplicial space]] [[!redirects fat geometric realization of a simplicial topological space]] [[!redirects fat geometric realization]] [[!redirects fat realization]]



\end{document}