\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*{simplicial skeleton}

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

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

[[!include homotopy - contents]]

\begin{quote}%
This entry is about the notion of (co)skeleta of [[simplicial sets]]. For the notion of skeleton of a [[category]] see at \emph{[[skeleton]]}.


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\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{compatibility_with_kan_conditions}{Compatibility with Kan conditions}\dotfill \pageref*{compatibility_with_kan_conditions} \linebreak
\noindent\hyperlink{Truncation}{Truncation and Postnikov towers}\dotfill \pageref*{Truncation} \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


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

For $\Delta$ the [[simplex category]] write $\Delta_{\leq n}$ for its [[full subcategory]] on the objects $[0], [1], \cdots, [n]$. The inclusion $\Delta|_{\leq n} \hookrightarrow \Delta$ induces a truncation functor

\begin{displaymath}
tr_n : sSet = [\Delta^{op}, Set] \to [\Delta_{\leq n}^{op},Set]
\end{displaymath}
that takes a [[simplicial set]] and restricts it to its degrees $\leq n$.

This functor has a [[left adjoint]], given by left [[Kan extension]]

\begin{displaymath}
sk_n 
  \;\colon\; 
  [\Delta_{\leq n},Set] \to  SSet
\end{displaymath}
called the $n$-\textbf{skeleton}

and a [[right adjoint]], given by right [[Kan extension]]

\begin{displaymath}
cosk_n 
  \;\colon\; 
  [\Delta_{\leq n},Set] \to SSet
\end{displaymath}
called the $n$-\textbf{coskeleton}.

\begin{displaymath}
( sk_n \dashv tr_n \dashv cosk_n) \;\;
   \colon
   \;\;
   sSet_{\leq n}
   \stackrel{\overset{sk_n}{\longrightarrow}}{\stackrel{\overset{tr_n}{\longleftarrow}}{\underset{cosk_n}{\longrightarrow}}}
   sSet
  \,.
\end{displaymath}
The $n$-skeleton produces a simplicial set that is freely filled with degenerate simplices above degree $n$.

Write

\begin{displaymath}
\mathbf{sk}_n := sk_n \circ tr_n: sSet \to sSet
\end{displaymath}
and

\begin{displaymath}
\mathbf{cosk}_n := cosk_n \circ tr_n: sSet \to sSet
\end{displaymath}
for the composite functors. Often by slight abuse of notation we suppress the boldface and just write $sk_n : sSet \to sSet$ and $cosk_n : sSet \to sSet$.

these in turn form an [[adjunction]]

\begin{displaymath}
( \mathbf{sk}_n \dashv \mathbf{cosk}_n)
  \;\;
  :
  \;\;
  sSet
  \stackrel{\leftarrow}{\to}
  sSet
  \,.
\end{displaymath}
So the $k$-coskeleton of a simplicial set $X$ is given by the formula

\begin{displaymath}
\mathbf{cosk}_k X : [n] \mapsto Hom_{sSet}(\mathbf{sk}_k \Delta[n], X)
  \,.
\end{displaymath}
Simplicial sets isomorphic to objects in the image of $cosk_n$ are called \textbf{$n$-coskeletal} simplicial sets.

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

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

\begin{prop}
\label{}\hypertarget{}{}
For $X \in$ [[sSet]], the following are equivalent:

\begin{itemize}%
\item $X$ is $n$-coskeletal;


\item on $X$ the unit $X \to \mathbf{cosk}_n(X)$ of the adjunction is an [[isomorphism]];


\item the map

\begin{displaymath}
X_k = Hom(\Delta[k], X) \stackrel{tr_n}{\to} Hom(tr_n(\Delta[k]), tr_n(X))
\end{displaymath}
is a bijection for all $k \gt n$


\item for $k \gt n$ and every morphism $\partial\Delta[k] \to X$ from the [[boundary of a simplex|boundary]] of the $k$-[[simplex]] there exists a \emph{unique} filler $\Delta[k] \to X$

\begin{displaymath}
\itexarray{
     \partial \Delta[k] &\to& X
     \\
     \downarrow & \nearrow
     \\
     \Delta[k]
  }
\end{displaymath}


\end{itemize}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
So in particular if $X$ is an $n$-coskeletal [[Kan complex]], all its [[simplicial homotopy group]]s above degree $(n-1)$ are trivial.

\end{remark}
\hypertarget{compatibility_with_kan_conditions}{}\subsubsection*{{Compatibility with Kan conditions}}\label{compatibility_with_kan_conditions}

\begin{prop}
\label{}\hypertarget{}{}
The coskeleton operations $\mathbf{cosk}_n$ preserve [[Kan complexes]].

$\mathbf{cosk}_n$ preserves those [[Kan fibrations]] between [[Kan complexes]] whose [[codomains]] have trivial [[homotopy group]] $\pi_n$.

\end{prop}
(\href{http://math.stackexchange.com/a/597990/58526}{Math.SE discussion})

\hypertarget{Truncation}{}\subsubsection*{{Truncation and Postnikov towers}}\label{Truncation}

\begin{prop}
\label{}\hypertarget{}{}
For each $n \in \mathbb{N}$, the [[unit of an adjunction|unit of the adjunction]]

\begin{displaymath}
X \to \mathbf{cosk}_n(X)
\end{displaymath}
induces an [[isomorphism]] on all [[simplicial homotopy groups]] in degree $\lt n$.

\end{prop}
It follows from the above that for $X$ a [[Kan complex]], the sequence

\begin{displaymath}
X = \underset{\leftarrow}{\lim}\, cosk_n X \to \cdots \to cosk_{n+1} X \to cosk_{n} X \to \cdots \to *
\end{displaymath}
is a [[Postnikov tower]] for $X$.

See also the discussion on \href{http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf#page=3}{p. 140, 141} of \href{http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf}{DwKan1984}.

For the interpretation of this in terms of [[(n,1)-topos]]es inside the [[(∞,1)-topos]] [[∞Grpd]] see [[n-truncated object in an (∞,1)-category]], example .

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

\begin{itemize}%
\item The [[nerve]] of a [[category]] is a 2-coskeletal simplicial set.


\item A [[Kan complex]] that is $(n+1)$-coskeletal is equivalent to (the [[nerve]] of) an [[n-groupoid]].


\item A 0-coskeletal simplicial set $X$ is (-1)-[[truncated]] and hence either empty or a [[contractible]] [[Kan complex]] , $X \stackrel{\simeq}{\to} *$ that is the [[nerve]] $X = N(C)$ of a [[groupoid]] $C$ that has a [[equivalence of categories]] $C \simeq *$.



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

\begin{itemize}%
\item [[Postnikov tower]], [[Whitehead tower]]


\item [[Eilenberg subcomplex]]


\item [[adjoint modality]]



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

Standard textbook references are

\begin{itemize}%
\item [[Peter May]], \emph{Simplicial methods in algebraic topology} (\href{http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu}{djvu})


\item [[Paul Goerss]] and [[Rick Jardine]], 1999, \emph{[[Simplicial homotopy theory]]}, number 174 in Progress in Mathematics, Birkh\"a{}user. (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{ps})



\end{itemize}
A classical article that amplifies the connection of the coskeleton operation to [[Postnikov towers]] is

\begin{itemize}%
\item [[William Dwyer]], [[Dan Kan]], \emph{An obstruction theory for diagrams of simplicial sets} (\href{http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf}{pdf})

\end{itemize}
The [[level of a topos]]-structure of simplicial (co-)skeleta is discussed in

\begin{itemize}%
\item C. Kennett, [[Emily Riehl|E. Riehl]], M. Roy, M. Zaks, \emph{Levels in the toposes of simplicial sets and cubical sets} , JPAA \textbf{215} no.5 (2011) pp.949-961. (\href{http://arxiv.org/abs/1003.5944}{arXiv:1003.5944})

\end{itemize}
[[!redirects coskeletal]] [[!redirects coskeleton]] [[!redirects simplicial coskeleton]]

[[!redirects coskeleta]] [[!redirects skeleta]]

[[!redirects simplicial skeleta]]



\end{document}