\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*{category of simplices}

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

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

[[!include category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{categories_of_simplices}{}\section*{{Categories of simplices}}\label{categories_of_simplices}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{colimits}{Colimits}\dotfill \pageref*{colimits} \linebreak
\noindent\hyperlink{the_nerve_and_subdivision}{The nerve and subdivision}\dotfill \pageref*{the_nerve_and_subdivision} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $X$ a [[simplicial set]] its \emph{category of simplices} is the [[category]] whose [[objects]] are the [[simplices]] \emph{in} $X$ and whose [[morphisms]] are maps between these, as simplices in $X$.

In particular the [[subcategory]] on the non-degenerate simplices has a useful interpretation: it is the [[poset]] of subsimplex inclusions whose [[nerve]] is the [[barycentric subdivision]] of $X$, at least if every non-degenerate simplex in $X$ comes from a monomorphism $\Delta^n \to X$, as for a [[simplicial complex]].

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

Let $X \in$ [[sSet]] be a [[simplicial set]].

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{category of simplices} of $X$ is equivalently (in increasing order of explicitness)

\begin{itemize}%
\item the [[category of elements]] of the [[presheaf]] $X_\bullet : \Delta^{op} \to Set$;


\item the [[comma category]] $(\Delta\downarrow X)$, where $\Delta$ denotes the [[Yoneda embedding]] $[n] \mapsto \Delta^n$.


\item the [[category]] whose [[objects]] are [[homomorphisms]] of simplicial sets $c : \Delta^n \to X$ from a standard simplicial [[simplex]] $\Delta^n$ to $X$, and whose morphisms $c \to c'$ are morphisms $f : \Delta^n \to \Delta^{n'}$ in the [[simplex category]] $\Delta$ such that the [[diagram]]

\begin{displaymath}
\itexarray{ 
    \Delta^n &&\stackrel{f}{\to}&& \Delta^{n'}
    \\
    & {}_{c}\searrow && \swarrow_{c'}  
    \\
    && X
  }
\end{displaymath}
[[commuting diagram|commutes]].



\end{itemize}
\end{defn}
\begin{defn}
\label{BarycentricSubdivision}\hypertarget{BarycentricSubdivision}{}
An $n$-[[simplex]] $x\in X_n$ is said to be \emph{non-degenerate} if it is not in the image of any degeneracy map.

Write

\begin{displaymath}
(\Delta\downarrow X)_{nondeg}\hookrightarrow (\Delta\downarrow X)
\end{displaymath}
for the (non-full) subcategory on the non-degenerate simplices. Notice that a morphism of $\Delta\downarrow X$ with source a non-degenerate simplex of $X$ is necessary a [[monomorphism]].

This is called the \textbf{category of non-degenerate simplices}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If every non-degenerate simplex in $X$ comes from a [[monomorphism]] $\Delta^n \to X$, then the [[nerve]] $N((\Delta \downarrow X)_{nondeg})$ is also called the \textbf{[[barycentric subdivision]]} of $X$.

\end{remark}
See at \emph{\href{subdivision#RelationToCategoryOfSimplices}{barycentric subdivision -- Relation to the category of simplices}}.

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

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

\begin{prop}
\label{}\hypertarget{}{}
If $X$ has the property that every face of every non-degenerate simplex is again non-degenerate, then the inclusion of the category of non-degenerate simplices $(\Delta \downarrow X)_{nondeg} \hookrightarrow (\Delta \downarrow X)$ has a [[left adjoint]] and is hence a [[reflective subcategory]].

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The category of simplices is a [[Reedy category]].

\end{prop}
\hypertarget{colimits}{}\subsubsection*{{Colimits}}\label{colimits}

Write $(\Delta \downarrow X) \to sSet$ for the canonical functor that sends $(\Delta^n \to X)$ to $\Delta^n$.

\begin{prop}
\label{ColimitOverSimplicesOfXIsX}\hypertarget{ColimitOverSimplicesOfXIsX}{}
The [[colimit]] over the [[functor]] $(\Delta \downarrow X) \to sSet$ is $X$ itself

\begin{displaymath}
X \simeq \underset{\to}{\lim}((\Delta \downarrow X) \to sSet)
\end{displaymath}
\end{prop}
\begin{proof}
By the [[co-Yoneda lemma]].

\end{proof}
In the textbook literature this appears for instance as (\hyperlink{Hovey}{Hovey, lemma 3.1.3}).

\begin{cor}
\label{}\hypertarget{}{}
A colimit-preserving [[functor]] $F\colon sSet \to C$ is uniquely determined by its action on the standard simplices:

\begin{displaymath}
F(X) \cong colim_{(\Delta\downarrow X)} F(\Delta^\bullet).
\end{displaymath}
\end{cor}
\begin{example}
\label{}\hypertarget{}{}
Important colimit-preserving functors out of [[sSet]] include

\begin{itemize}%
\item [[geometric realization]] of simplicial sets
\item [[subdivision]] of simplicial sets
\item the [[left adjoint]] to the [[nerve]] functor $N:Cat \to SSet$
\item the category of simplices itself, as a functor $SSet \to Cat$; see \href{category+of+elements#ColimitPreserving}{category of elements -- colimit preserving}.

\end{itemize}
\end{example}
\hypertarget{the_nerve_and_subdivision}{}\subsubsection*{{The nerve and subdivision}}\label{the_nerve_and_subdivision}

Let $N\colon$ [[Cat]] $\to$ [[sSet]] denote the simplicial [[nerve]] functor on [[categories]].

\begin{theorem}
\label{}\hypertarget{}{}
The functor $sSet \to sSet$ that assigns [[barycentric subdivision]], def. \ref{BarycentricSubdivision},

\begin{displaymath}
X\mapsto N(\Delta\downarrow X)
\end{displaymath}
preserves [[colimits]].

\end{theorem}
\begin{proof}
An $n$-[[simplex]] of $N(\Delta\downarrow X)$ is determined by a string of $n+1$ composable morphisms

\begin{displaymath}
\Delta^{k_n} \to \dots\to \Delta^{k_0}
\end{displaymath}
along with a map $\Delta^{k_0} \to X$, i.e. an element of $X_{k_0}$ Thus, each the functor $X\mapsto N(\Delta\downarrow X)_n$ from $SSet \to Set$ is a coproduct of a family of ``evaluation'' functors. Since evaluation preserve colimits, coproducts commute with colimits, and colimits in $SSet$ are levelwise, the statement follows.

\end{proof}
Therefore, the simplicial set $N(\Delta\downarrow X)$ itself can be computed as a colimit over the category $(\Delta\downarrow X)$ of the simplicial sets $N(\Delta\downarrow \Delta^n)$.

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

A basic disussion is for instance in section 3.1 of

\begin{itemize}%
\item [[Mark Hovey]], \emph{Model categories}, Mathematical surveys and monographs volume 63, American Mathematical Society

\end{itemize}
Homotopy finality of the non-degenerate simplices is discussed in section 4.1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
For more on [[barycentric subdivision]] see also section 2 of

\begin{itemize}%
\item [[Rick Jardine]], \emph{Simplicial approximation}, Theory and Applications of Categories, Vol. 12, 2004, No. 2, pp 34-72. (\href{ftp://ftp.math.ethz.ch/EMIS/journals/TAC/volumes/12/2/12-02aabs.html}{web})

\end{itemize}
[[!redirects category of simplices]] [[!redirects category of simplicies]] [[!redirects categories of simplices]]

[[!redirects category of nondegenerate simplices]] [[!redirects categories of nondegenerate simplices]] [[!redirects category of non-degenerate simplices]] [[!redirects categories of non-degenerate simplices]]



\end{document}