\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*{Quillen bifunctor}

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

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

[[!include model category theory - 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{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{monoidal_and_enriched_model_categories}{Monoidal and enriched model categories}\dotfill \pageref*{monoidal_and_enriched_model_categories} \linebreak
\noindent\hyperlink{CoendsOverTensors}{Lift to coends over tensors}\dotfill \pageref*{CoendsOverTensors} \linebreak
\noindent\hyperlink{bousfieldkan_type_homotopy_colimits}{Bousfield-Kan type homotopy colimits}\dotfill \pageref*{bousfieldkan_type_homotopy_colimits} \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}

A \emph{(left) Quillen [[bifunctor]]} is a [[functor]] of two variables between [[model category|model categories]] that respects combined [[cofibrations]] in its two arguments in a suitable sense.

The notion of Quillen bifunctor enters the definition of [[monoidal model category]] and of [[enriched model category]].

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

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(Quillen bifunctor)}

Let $C, D, E$ be [[model category|model categories]]. A [[functor]] $F : C \times D \to E$ is a \textbf{Quillen bifunctor} if it satisfies the following two conditions:

\begin{enumerate}%
\item for any cofibration $i : c \to c'$ in $C$ and cofibration $j : d \to d'$ in $D$, the induced ([[pushout product]]) morphism

\begin{displaymath}
F(c', d) \coprod_{F(c,d)} F(c,d') \to F(c', d')
\end{displaymath}
is a cofibration in $E$, which is a weak equivalence if either $i$ or $j$ is a weak equivalence


\item it preserves [[colimit]]s separately in each variable



\end{enumerate}
\end{defn}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

In full detail the [[pushout]] appearing in the first condition is the one sitting in the pushout diagram

\begin{displaymath}
\itexarray{
    F(c,d) &\stackrel{F(Id,j)}{\to}& F(c,d')
    \\
    \;\;\downarrow^{F(i,Id)} && \downarrow
    \\
    F(c',d) &\stackrel{}{\to}&
    F(c', d) \coprod_{F(c,d)} F(c,d') 
  }
  \,.
\end{displaymath}
In particular, if $i = (\emptyset \hookrightarrow c)$ we have $F(\emptyset, d) = F(\emptyset, d') = \emptyset$ (since the [[initial object]] is the [[colimit]] over the empty diagram and $F$ is assumed to preserve colimits) and the above pushout diagram reduces to

\begin{displaymath}
\itexarray{
    \emptyset &{\to}& \emptyset
    \\
    \;\;\downarrow && \downarrow
    \\
    F(c,d) &\stackrel{}{\to}&
    F(c,d) 
  }
  \,.
\end{displaymath}
Therefore for $c$ a cofibrant object the condition is that $F(c,-) : D \to E$ preserves cofibrations and acyclic cofibrations. Similarly for $d$ fibrant the condition is that $F(-,d) : C \to E$ preserves cofibrations and acyclic cofibrations.

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

\begin{uprop}
Let $\otimes : C \times D \to E$ be an [[adjunction of two variables]] between model categories and assume that $C$ and $D$ are [[cofibrantly generated model categories]]. Then $\otimes$ is a Quillen bifunctor precisely if it satisfies its axioms on generating (acyclic) cofibrations, i.e. if for $f : c_1 \to c_2$ and $g : d_1 \to d_2$ we have for the morphism

\begin{displaymath}
(c_1 \otimes d_2) \coprod_{c_1 \otimes d_1} (c_2 \otimes d_1)
  \to
  c_2 \otimes d_2
\end{displaymath}
is

\begin{itemize}%
\item a cofibration if both $f$ and $g$ are generating cofibrations;


\item an acyclic cofibration if one is a generating cofibration and the other a generating acyclic cofibration.



\end{itemize}
\end{uprop}
This appears for instance as Corollary 4.2.5 in

\begin{itemize}%
\item [[Mark Hovey]], \emph{Model Categories}

\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{monoidal_and_enriched_model_categories}{}\subsubsection*{{Monoidal and enriched model categories}}\label{monoidal_and_enriched_model_categories}

\begin{itemize}%
\item In a [[monoidal model category]] $C$ the [[tensor product]] $\otimes : C \times C \to C$ is required to be a Quillen bifunctor.


\item An [[enriched model category]] $D$ over the [[monoidal model category]] $C$ is one that is [[power]]ed and [[copower]]ed over $D$ such that the [[copower]] $\otimes : D \times C \to D$ is a Quillen bifunctor.



\end{itemize}
\hypertarget{CoendsOverTensors}{}\subsubsection*{{Lift to coends over tensors}}\label{CoendsOverTensors}

The following proposition asserts that under mild conditions a Quillen bifunctor on $C \times D$ lifts to a Quillen bifunctor on [[functor category|functor categories]] of functors to $C$ and $D$.

\begin{uprop}
Let $\otimes : C \times D \to E$ be a Quillen functor. Let

\begin{itemize}%
\item $S$ be a [[Reedy category]] and take the [[functor category|functor categories]] $[S,C]$ and $[S^{op},C]$ be equipped with the corresponding[[Reedy model structure]].


\item \emph{or} assume that $C$ and $D$ are [[combinatorial model category|combinatorial model categories]] and let $[S,C]$ and $[S^{op},A]$ be equipped, respectively with the projective and the injective globel [[global model structure on functors|model structure on functor categories]].



\end{itemize}
Then the [[coend]] [[functor]]

\begin{displaymath}
\int^{S} (- \otimes -) : [S,C]\times [S^{op},D] \to E
\end{displaymath}
is again a 'Quillen bifunctor.

\end{uprop}
This \hyperlink{Lurie}{Lurie, prop. A.2.9.26 with remark A.2.9.27}.

It follows that the corresponding left [[derived functor]] computes the corresponding [[homotopy coend]].

\hypertarget{bousfieldkan_type_homotopy_colimits}{}\subsubsection*{{Bousfield-Kan type homotopy colimits}}\label{bousfieldkan_type_homotopy_colimits}

This is an application of the above application.

Let $C$ be a [[category]] and $A$ be a [[simplicial model category]]. Let $F : C \to A$ be a functor and let ${*} : C^{op} \to A$ be the functor constant on the terminal object.

Consider the [[global model structure on functors]] $[C^{op},SSet]_{proj}$ and $[C^{op},A]_{inj}$ and let $Q({*})_{proj}$ be a cofibrant replacement for ${*}$ in $[C^{op},Set]_{proj}$ and $Q_{inj}(F)$ a cofibrant replacement for $F$ in $[C,A]_{inj}$.

One show that the [[homotopy colimit]] over $F$ is computed as the [[coend]] or [[weighted limit]]

\begin{displaymath}
hocolim F = \int Q_{proj}({*}) \cdot Q_{inj}(F)
  \,.
\end{displaymath}
One possible choice is

\begin{displaymath}
Q_{proj}({*})
  =
  N(-/C)^{op}
  \,.
\end{displaymath}
That this is indeed a projectively cofibrant resulution of the constant on the [[terminal object]] is for instance proposition 14.8.9 of

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

\end{itemize}
For the case that $C = \Delta^{op}$ this is the classical choice by Bousfield and Kan, see [[Bousfield-Kan map]].

Assume that $A$ takes values in cofibrant objects of $A$, then it is already cofibrant in the injective model structure $[C,A]_{inj}$ on functors and we can take $Q_{inj}(F) = F$. Then the above says that

\begin{displaymath}
hocolim F = \int N(-/C)^\op \cdot F
  \,.
\end{displaymath}
For $C = \Delta$ this is the classical prescription by Bousfield-Kan for homotopy colimits, see also the discussion at [[weighted limit]].

Using the above proposition, it follows in particular explicitly that the homotopy colimit preserves degreewise cofibrations of functors over which it is taken.

A nice discussion of this is in (\hyperlink{Gambino}{Gambino}).

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

\begin{itemize}%
\item [[two-variable adjunction]]

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

Appendix A.2 of

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


\item [[Nicola Gambino]], \emph{Weighted limits in simplicial homotopy theory} (\href{http://www.crm.cat/Publications/08/Pr790.pdf}{pdf})



\end{itemize}


\end{document}