\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*{(infinity,1)-categorical hom-space}

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

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

[[!include homotopy - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\hypertarget{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space}

[[!include mapping space - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{presentations}{Presentations}\dotfill \pageref*{presentations} \linebreak
\noindent\hyperlink{ForACategoryWithWeakEquivalences}{For a category with weak equivalences}\dotfill \pageref*{ForACategoryWithWeakEquivalences} \linebreak
\noindent\hyperlink{Framings}{For a model category}\dotfill \pageref*{Framings} \linebreak
\noindent\hyperlink{EnrichedHomsCofToFib}{For a simplicial model category}\dotfill \pageref*{EnrichedHomsCofToFib} \linebreak
\noindent\hyperlink{comparison}{Comparison}\dotfill \pageref*{comparison} \linebreak
\noindent\hyperlink{InACategoryOfFibrantObjects}{In a category of fibrant objects}\dotfill \pageref*{InACategoryOfFibrantObjects} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SpacesOfEquivalences}{Hom-spaces of equivalences}\dotfill \pageref*{SpacesOfEquivalences} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Where an ordinary [[category]] has a [[hom-set]], an [[(∞,1)-category]] has an [[∞-groupoid]] of morphisms between any two objects, a \emph{hom-space}.

There are several ways to \emph{present} an [[(∞,1)-category]] $\mathbf{C}$ by an ordinary [[category]] $C$ equipped with some extra structure: for instance $C$ may be a [[category with weak equivalences]] or a [[model category]] or even a [[simplicial model category]]. In all of these presentations, given two objects $X, Y \in C$, there is a way to construct a [[simplicial sets]] $\mathbb{R}\mathbf{C}(X,Y)$ that presents the hom-[[∞-groupoid]] $\mathbf{C}(X,Y)$. This simplicial set -- or rather its [[homotopy type]] -- is called the \emph{derived hom space} or \emph{homotopy function complex} and denoted $\mathbf{R}Hom(X,Y)$ or similarly.

\hypertarget{presentations}{}\subsection*{{Presentations}}\label{presentations}

There are many ways to present an [[(∞,1)-category]] by [[category theory|category theoretic data]], and for each of these there are corresponding tools for explicitly computing the derived hom spaces.

The most basic data is that of a [[category with weak equivalences]]. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called \emph{[[simplicial localization]]}. This we discuss below in \emph{\hyperlink{ForACategoryWithWeakEquivalences}{For a category with weak equivalences}}.

Particularly useful extra structure on a [[category with weak equivalences]] that helps with computing the derived hom spaces is the structure of a \emph{[[model category]]}. Using this one can construct simplicial resolutions of objects -- called \emph{framings} -- that generalize [[cylinder objects]] and [[path objects]], and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in \emph{\hyperlink{Framings}{For a model category}}.

Still a bit more helpful structure on top of a bare model category is that of a [[simplicial model category]]. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the [[sSet]]-[[hom objects]]. This we discuss below in \emph{\hyperlink{EnrichedHomsCofToFib}{For a simplicial model category}}.

\hypertarget{ForACategoryWithWeakEquivalences}{}\subsubsection*{{For a category with weak equivalences}}\label{ForACategoryWithWeakEquivalences}

Let $(C,W \subset Mor(C))$ be a [[category with weak equivalences]].

\begin{defn}
\label{ZigZagCategories}\hypertarget{ZigZagCategories}{}
Fix $n \in \mathbb{N}$. For $X,Y \in Obj(C)$, define a category $wMor_C^n(X,Y)$

\begin{itemize}%
\item whose objects are [[zig-zag]]s of morphisms in $C$ of length $n$

\begin{displaymath}
X = X_0 \leftarrow X_1 \to X_2 \leftarrow \cdots \to X_{n-1} \leftarrow X_n = Y
\end{displaymath}
such that each morphism going to the left, $X_{2k}\leftarrow X_{2k +1}$, is a [[weak equivalence]], an element in $W$;


\item morphisms between such objects $(X,X_i,Y) \to (X',X'_i,Y')$ are collections of weak equivalences $(X_i \to X'_i)$ for all $0 \lt i \lt n$ such that all triangles and squares commute.



\end{itemize}
\end{defn}
\begin{defn}
\label{HammockLocalization}\hypertarget{HammockLocalization}{}
Write $N(wMor_C^n(X,Y))$ for the [[nerve]] of this category, a [[simplicial set]].

The \emph{[[hammock localization]]} $L_W^H C$ of $C$ is the [[simplicially enriched category]] with objects those of $C$ and [[hom-objects]] given by the [[colimit]] over the length of these hammock hom-categories

\begin{displaymath}
L^H C(X,Y) := \lim_{\to_n} N(wMor_C^n(X,Y))
  \,.
\end{displaymath}
The [[Kan fibrant replacement]] of this simplicial set is the derived hom-space between $X$ and $Y$ of the $(\infty,1)$-category modeled by $(C,W)$.

\end{defn}
\hypertarget{Framings}{}\subsubsection*{{For a model category}}\label{Framings}

The derived hom spaces of a model category $C$ may always be computed in terms of simplicial resolutions with respect to the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$. These resolutions are often called \emph{framings} (\hyperlink{Hovey}{Hovey}). These constructions are originally due to (\hyperlink{DHK}{Dwyer-Hirschhorn-Kan}).

Let $C$ be any [[model category]].

\begin{prop}
\label{}\hypertarget{}{}
There is an [[adjoint triple]]

\begin{displaymath}
(const \dashv ev_0 \dashv (-)^{\times^\bullet})
  : 
  C
  \stackrel{\overset{const}{\longrightarrow}}{\stackrel{\overset{ev_0}{\longleftarrow}}{\underset{(-)^{\times^\bullet}}{\longrightarrow}}}
  \,,
  [\Delta^{op}, C]
  \,,
\end{displaymath}
where

\begin{enumerate}%
\item $const X : [n] \mapsto X$;


\item $ev_0 X_\bullet = X_0$;


\item $X^{\times^\bullet} : [n] \mapsto X^{\times^{n+1}}$.



\end{enumerate}
\end{prop}
\begin{remark}
\label{CoDiscreteIsReedyFibrant}\hypertarget{CoDiscreteIsReedyFibrant}{}
For $X \in C$ fibrant, $X^{\times^\bullet}$ is fibrant in the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.

\end{remark}
\begin{proof}
The matching morphisms are in fact [[isomorphisms]].

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a model category.

\begin{enumerate}%
\item For $X \in C$ any object, a \emph{simplicial frame} on $X$ is a factorization of $const X \to X^{\times^\bullet}$ into a weak equivalence followed by a fibration in the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.


\item A \emph{right framing} in $C$ is a functor $(-)_\bullet :  C \to [\Delta^{op}, C]$ with a [[natural isomorphism]] $(X)_0 \simeq X$ such that $X_\bullet$ is a simplicial frame on $X$.



\end{enumerate}
Dually for \emph{cosimplicial frames}.

\end{defn}
This appears as (\hyperlink{Hovey}{Hovey, def. 5.2.7}).

\begin{remark}
\label{}\hypertarget{}{}
By remark \ref{CoDiscreteIsReedyFibrant} a simplicial frame $X_\bullet$ in the above is in particular fibrant in $[\Delta^{op}, C]_{Reedy}$.

\end{remark}
\begin{prop}
\label{SimplicialFunctionComplexes}\hypertarget{SimplicialFunctionComplexes}{}
For $X \in C$ cofibrant and $A \in C$ fibrant, there are weak equivalences in $sSet_{Quillen}$

\begin{displaymath}
Hom_C(X^\bullet, A)
  \stackrel{\simeq}{\to}
  diag Hom_C(X^\bullet, A_\bullet)
  \stackrel{\simeq}{\leftarrow}
  Hom_C(X, A_\bullet)
  \,,
\end{displaymath}
(where in the middle we have the diagonal of the [[bisimplicial set]] $Hom(X^\bullet, A_\bullet)$).

\end{prop}
This appears as (\hyperlink{Hovey}{Hovey, prop. 5.4.7}).

Either of these simplicial sets is a model for the derived hom-space $\mathbb{R}Hom(X,A)$.

\begin{remark}
\label{}\hypertarget{}{}
By developing these constructions further, one obtains a canonical [[simplicial model category]]-resolution of (left proper and combinatorial) model categories $C$, such that the simplicial resolutions given by framings are just the cofibrant$\to$fibrant $sSet$-hom objects as discussed \hyperlink{EnrichedHomsCofToFib}{below}.

This is discussed at \emph{\href{http://ncatlab.org/nlab/show/simplicial+model+category#SimpEquivMods}{Simplicial Quillen equivalent models}}.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be a model category, let $\mathrm{c}_\mathrm{w} C$ be the full subcategory of $[\Delta, C]$ spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let $\mathrm{s}_\mathrm{w} C$ be the full subcategory of $[\Delta^{op}, C]$ spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.

\begin{enumerate}%
\item $const : C \to \mathrm{c}_\mathrm{w} C$ is the right half of an adjoint homotopical equivalence of [[homotopical category|homotopical categories]], and $const : C \to \mathrm{s}_\mathrm{w} C$ is the left half of an adjoint homotopical equivalence of homotopical categories.
\item The functor $\operatorname{diag} Hom_C : (\mathrm{c}_\mathrm{w} C)^{op} \times \mathrm{s}_\mathrm{w} C \to sSet$ admits a right [[derived functor]].
\item The induced functor $(\operatorname{Ho} C)^{op} \times \operatorname{Ho} C \to \operatorname{Ho} sSet$ is the derived hom-space functor.

\end{enumerate}
\end{prop}
\hypertarget{EnrichedHomsCofToFib}{}\subsubsection*{{For a simplicial model category}}\label{EnrichedHomsCofToFib}

We describe here in more detail properties of [[derived hom-functors]] (see there for more) in a [[simplicial model category]].

The crucial axiom used for this is the axiom of an [[enriched model category]] $C$ which says that

\begin{itemize}%
\item the [[copower|tensor operation]]

\begin{displaymath}
\cdot : C \times SSet \to C
\end{displaymath}
is a [[Quillen bifunctor]];


\item or equivalently that for $X \to Y$ a cofibration and $A \to B$ a fibration the induced morphism

\begin{displaymath}
C(Y, A) \to C(X,A) \times_{C(X,B)} C(Y,B)
\end{displaymath}
is a fibration, which is acyclic if either $X \to Y$ or $A \to B$ is.



\end{itemize}
First of all the first statement directly implies that for $\emptyset \in C$ the [[initial object]] and $A \in C$ any object, the simplicial set $C(\emptyset,A) = {*}$ is the terminal simplicial set (see also \href{powered+and+copowered+category#InTensoredCotensoredCategoryInitialObjectIsEnrichedInitial}{this Prop.}): because for any simplicial set $S$

\begin{displaymath}
\begin{aligned}
    SSet(S,C(\emptyset, A))
    & =  Hom_C(\emptyset \cdot S, A)
    \\
    & = Hom_C(colim_{\emptyset} \cdot S, A)
    \\
    & = Hom_C(\emptyset, A)
    \\
    &= {*}
  \end{aligned}
  \,,
\end{displaymath}
where we use that the [[copower|tensor]] [[Quillen bifunctor]] is required to respect [[colimit]]s and that the empty colimit is the [[initial object]]. (All equality signs here denote [[isomorphisms]], to distinguish them from weak equivalences.)

Similarly one has for all $X$ that $C(X,{*}) = {*}$.

Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.

\begin{lemma}
\label{}\hypertarget{}{}
In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $A \in C$ fibrant, the [[simplicial set]] $C(X,A)$ is a [[Kan complex]].

\end{lemma}
\begin{proof}
We apply the [[enriched model category]] axiom to the cofibration $\emptyset \to X$ and the fibration $A \to {*}$ to obtain a fibration

\begin{displaymath}
C(X,A) \to C(\emptyset, A) \times_{C(\emptyset,{*})} C(X,{*})  
  \,.
\end{displaymath}
The right hand is the [[pullback]] of the terminal simplicial set ${*} = \Delta^0$ to itself, hence is itself the point. So we have a fibration $C(X,A) \to {*}$ and $C(X,A)$ is a fibrant object in the standard [[model structure on simplicial sets]], hence a [[Kan complex]]. .

\end{proof}
\begin{lemma}
\label{}\hypertarget{}{}
In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $f : A \to B$ a fibration, the morphism of [[simplicial set]]s $C(X,f) : C(X,A) \to C(X,B)$ is a [[Kan fibration]] that is a [[weak homotopy equivalence]] if $f$ is acyclic.

Dually, for $i : X \to Y$ a cofibration and $A$ fibrant, the morphism $C(i,A) : C(X,A) \to C(Y,A)$ is a cofibration of simplicial sets.

\end{lemma}
\begin{proof}
This is as before. Explicity, consider the first case, the second one is the formal dual of that:

We enter the enrichment axiom with the morphisms $\emptyset \to X$ and $A \to B$ and find for the required [[pullback]] that

\begin{displaymath}
C(\emptyset,A) \times_{C(\emptyset, B)} C(X,B)
  =
  {*} \times_{*} C(X,B)
  =
  C(X,B)
\end{displaymath}
and hence that $C(X,A) \to C(X,B)$ is, indeed, a fibration, which is acyclic if $A \to B$ is.

\end{proof}
\begin{proposition}
\label{}\hypertarget{}{}
Let $C$ be a [[simplicial model category]].

Then for $X$ a cofibant object and

\begin{displaymath}
f : A \stackrel{\simeq}{\to}  B
\end{displaymath}
a weak equivalence between fibrant objects, the [[enriched functor|enriched]] [[hom-object|hom-functor]]

\begin{displaymath}
C(X,f) : C(X,A) \to C(X,B)
\end{displaymath}
is a [[weak homotopy equivalence]] of [[Kan complex]]es.

Similarly, for $A$ a fibrant object and $j : X \stackrel{\simeq}{\to} Y$ a weak equivalence between cofibrant objects, the morphism

\begin{displaymath}
C(j,A) : C(X,A) \to C(Y,A)
\end{displaymath}
is a [[weak homotopy equivalence]] of [[Kan complex]]es.

\end{proposition}
\begin{proof}
The second case is formally dual to the first, so we restrict attention to the first one.

By the above, the axioms of an [[enriched model category]] ensure that the above statement is true when $f$ is in addition a fibration. So we reduce the situation to that case.

This is possible because both $A$ and $B$ are assumed to be fibrant. This allows to apply the \emph{factorization lemma} that is described in great detail at [[category of fibrant objects]]. By this lemma, for every morphism $f : A \to B$ between fibrant objects there is a commutative diagram

\begin{displaymath}
\itexarray{
    && \mathbf{E}_f B
    \\
    & {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}}
    \\
    A &&\stackrel{\simeq}{\to}&& B
  }
\end{displaymath}
Since $f$ is assumed a weak equivalence it follows by [[category with weak equivalences|2-out-of-3]] that $\mathbf{E}_f B$ is also a weak equivalence.

Therefore by the above properties of simpliciall enriched categories we obtain a [[span]] of acyclic fibrations of [[Kan complex]]es

\begin{displaymath}
C(X,A) \stackrel{\simeq}{\leftarrow}
  C(X, \mathbf{E}_f B)
  \stackrel{\simeq}{\to} 
  C(X,B)
  \,.
\end{displaymath}
By the [[Whitehead theorem]] every weak equivalence of Kan complexes is a [[homotopy equivalence]], hence there is a weak equivalence

\begin{displaymath}
C(X,A) \stackrel{\simeq}{\to} C(X,\mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B)
\end{displaymath}
that is homotopic to our $C(X,f)$. Therefore this is also a weak equivalence.

\end{proof}
\hypertarget{comparison}{}\subsubsection*{{Comparison}}\label{comparison}

Let $C$ be a [[model category]]. We discuss how its simplicial function complexes from prop. \ref{SimplicialFunctionComplexes} are related to the simplicial localization from def. \ref{ZigZagCategories} and def. \ref{HammockLocalization}.

Suppose now that $Q : C \to C$ is a [[cofibrant replacement functor]] and $R : C \to C$ a [[fibrant replacement functor]], $\Gamma^\bullet : C \to (cC)_c$ a [[cosimplicial resolution functor]] and $\Lambda_\bullet : C \to (sC)_f$ a [[simplicial resolution functor]] in the [[model category]] $C$.

\begin{theorem}
\label{DKTheorem}\hypertarget{DKTheorem}{}
\textbf{(Dwyer--Kan)}

There are natural weak equivalences between the following equivalent realizations of this [[SSet]] [[hom-object]]:

\begin{displaymath}
\itexarray{
    Mor_C(\Gamma^\bullet X, R Y)
    &\stackrel{\simeq}{\to}&
    diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y)
    &\stackrel{\simeq}{\leftarrow}&
    Mor_C(Q X, \Lambda_\bullet Y)
    \\
    &&
    \uparrow^\simeq
    \\
    &&
    hocolim_{p,q \in \Delta^{op} \times \Delta^{op}}
    Mor_C(\Gamma^p X, \Lambda_q Y)
    \\
    &&\downarrow^\simeq
    \\
    &&N wMor_C^3(X,Y)
    \\
    &&\downarrow^\simeq
    \\
    &&Mor_{L^H C}(X,Y)
  }
  \,.
\end{displaymath}
\end{theorem}
The top row weak equivalences are those of prop. \ref{SimplicialFunctionComplexes}

\hypertarget{InACategoryOfFibrantObjects}{}\subsubsection*{{In a category of fibrant objects}}\label{InACategoryOfFibrantObjects}

There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a [[category of fibrant objects]]. This is described in (\hyperlink{Cisinski10}{Cisinksi 10}) and (\hyperlink{NSS12}{Nikolaus-Schreiber-Stevenson 12, section 3.6.2}).

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

\hypertarget{SpacesOfEquivalences}{}\subsubsection*{{Hom-spaces of equivalences}}\label{SpacesOfEquivalences}

\begin{theorem}
\label{DKTheorem}\hypertarget{DKTheorem}{}
For $C$ a [[simplicial model category]] and $X$ an object, the [[delooping]] of the [[automorphism ∞-group]]

\begin{displaymath}
Aut_W(X) \subset \mathbb{R}Hom(X,X)
\end{displaymath}
has the [[homotopy type]] of the component on $X$ of the [[nerve]] $N(C_W)$ of the [[subcategory]] of weak equivalences:

\begin{displaymath}
\mathbf{B} Aut_W(X) \simeq N(C_W)_X
  \,.
\end{displaymath}
The equivalence is given by a finite sequence of [[zig-zags]] and is natural with respect to [[sSet]]-[[enriched functors]] of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for $X$ again to a fibrant cofibrant object.

\end{theorem}
This is \hyperlink{DK84}{Dwyer-Kan 84, 2.3, 2.4}.

\begin{cor}
\label{}\hypertarget{}{}
For $C$ a model category, the simplicial set $N(C_W)$ is a model for the [[core]] of the [[(∞,1)-category]] determined by $C$.

\end{cor}
\begin{proof}
That core, like every [[∞-groupoid]] is equivalent to the disjoint union over its connected components of the deloopings of the automorphism $\infty$-groups of any representatives in each connected component.

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

\begin{itemize}%
\item [[hom-object]]


\item [[hom-set]], [[hom-functor]]


\item [[hom-category]]


\item [[hom-space]], \textbf{derived hom-space}, [[cocycle space]]



\end{itemize}
[[!include homotopy-homology-cohomology]]

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

For some original references by [[William Dwyer]] and [[Dan Kan]] see [[simplicial localization]]. For instance

\begin{itemize}%
\item [[William Dwyer]], [[Dan Kan]], \emph{A classication theorem for diagrams of simplicial sets}, Topology 23 (1984), 139-155.

\end{itemize}
The theory of \emph{framings} is due to

\begin{itemize}%
\item [[William Dwyer]], [[Philip Hirschhorn]], [[Dan Kan]], \emph{Model categories and general abstract homotopy theory}, (1997) (\href{http://www.mimuw.edu.pl/~jacho/literatura/ModelCategory/DHK_ModelCateogories1.pdf}{pdf})

\end{itemize}
and in parallel section 5 of

\begin{itemize}%
\item [[Mark Hovey]], \emph{Model categories} (\href{http://math.unice.fr/~brunov/SecretPassage/Hovey-Model%20Categories.ps}{ps})

\end{itemize}
and in sections 16, 17 of

\begin{itemize}%
\item [[Philip Hirschhorn]], \emph{Model categories and their localization} .

\end{itemize}
A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of

\begin{itemize}%
\item [[Clark Barwick]], \emph{On (enriched) left Bousfield localization of model categories} (\href{http://arxiv.org/abs/0708.2067}{arXiv})

\end{itemize}
Discussion of derived hom spaces for [[categories of fibrant objects]] is in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Invariance de la K-th\'e{}orie par equivalences d\'e{}riv\'e{}es}, J. K-theory, 6 (2010), 505--546.

\end{itemize}
and section 3.6.2 of

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]} (\href{http://arxiv.org/abs/1207.0249}{arXiv:1207.0249})

\end{itemize}
[[!redirects (infinity,1)-categorical hom-spaces]]

[[!redirects (∞,1)-categorical hom-space]] [[!redirects (∞,1)-categorical hom-spaces]]

[[!redirects (infinity,1)-categorical hom space]] [[!redirects (∞,1)-categorical hom space]] [[!redirects (infinity,1)-categorial hom-space]] [[!redirects (∞,1)-categorial hom-space]] [[!redirects (infinity,1)-categorial hom space]] [[!redirects (∞,1)-categorial hom space]]

[[!redirects derived hom space]] [[!redirects derived hom-space]] [[!redirects derived hom spaces]] [[!redirects derived hom-spaces]]

[[!redirects hom-space]] [[!redirects hom-spaces]]

[[!redirects hom-∞-groupoid]] [[!redirects hom-infinity-groupoid]] [[!redirects hom-(∞,0)-category]] [[!redirects hom-(infinity,0)-category]] [[!redirects hom ∞-groupoid]] [[!redirects hom infinity-groupoid]] [[!redirects hom (∞,0)-category]] [[!redirects hom (infinity,0)-category]]

[[!redirects homotopy function complex]] [[!redirects homotopy function complexes]]

[[!redirects (∞,1)-hom (∞,1)-functor)]]

[[!redirects (∞,1)-categorical hom]]

[[!redirects (∞,1)-categorical hom-spaces]]



\end{document}