\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*{coend in a derivator}

\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{ends_and_coends_in_a_derivator}{}\section*{{Ends and coends in a derivator}}\label{ends_and_coends_in_a_derivator}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{as_a_colimit_over_simplices}{As a colimit over simplices}\dotfill \pageref*{as_a_colimit_over_simplices} \linebreak
\noindent\hyperlink{as_a_colimit_over_arrows}{As a colimit over arrows}\dotfill \pageref*{as_a_colimit_over_arrows} \linebreak
\noindent\hyperlink{as_a_bar_construction}{As a bar construction}\dotfill \pageref*{as_a_bar_construction} \linebreak
\noindent\hyperlink{as_a_weighted_colimit}{As a weighted colimit}\dotfill \pageref*{as_a_weighted_colimit} \linebreak


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

[[end|Ends]] and [[coends]] are special sorts of [[limit]] and [[colimit]], respectively, and have corresponding sorts of [[homotopy limits]] and colimits -- [[homotopy ends]] and coends. Since a [[derivator]] is a formal structure for computing homotopy limits and colimits, there are corresponding notions of ends and coends in a derivator.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $D$ be a [[derivator]] indexed on a 2-category $Dia$ of diagram shapes, let $A\in Dia$, and let $H\in D(A^{op}\times A)$; we wish to define the \emph{coend} of $H$ (obvious dualizations will yield its \emph{end}). We will give four equivalent definitions, each of which generalizes a classical construction of ordinary coends in terms of colimits.

\hypertarget{as_a_colimit_over_simplices}{}\subsubsection*{{As a colimit over simplices}}\label{as_a_colimit_over_simplices}

One classical construction of a coend as a colimit proceeds by constructing an auxiliary category $A^{tw}$, whose objects are the objects and arrows of $A$, with morphisms from each arrow of $A$ (regarded as an object of $A^{tw}$) to its domain and codomain. There is a functor $p\colon A^{tw}\to A^{op}\times A$ which sends each object $x$ to $(x,x)$ and each arrow $f\colon x\to y$ to $(y,x)$, and the coend of $H\colon A^{op}\times A\to C$ can be constructed as the colimit of $p^* H\colon A^{tw}\to C$.

We can mimic this in a derivator, except that we need to ``homotopify'' $A^{tw}$ by including higher information as well. Thus, let $(\Delta / A)^{op}$ denote the opposite of the category of simplices of $A$. Thus its objects are functors $x\colon [n]\to A$, and its morphisms from $x\colon [n]\to A$ to $y\colon [m]\to A$ are functors $[m]\to [n]$ making the evident triangle commute. There is a functor $p\colon (\Delta / A)^{op} \to A^{op}\times A$ which sends $x\colon [n]\to A$ to $(x(n), x(0))$.

Note that $A^{tw}$, as constructed above, is the full subcategory of $(\Delta / A)^{op}$ containing only the 0-simplices and 1-simplicies. The inclusion of this subcategory is [[final functor|final]], but not [[homotopy final functor|homotopy final]]. Thus, for ordinary colimits it suffices to consider $A^{tw}$, but for homotopy colimits we need all of $(\Delta / A)^{op}$.

Hence, we define the \textbf{(homotopy) coend} of $H\in D(A^{op}\times A)$ to be the (homotopy) colimit of $p^* H \in D((\Delta / A)^{op})$.

\hypertarget{as_a_colimit_over_arrows}{}\subsubsection*{{As a colimit over arrows}}\label{as_a_colimit_over_arrows}

Another classical construction of a coend as a colimit involves a different auxiliary category, the opposite [[twisted arrow category]] $Tw(A)^{op}$. The objects of $Tw(A)^{op}$ are arrows $f\colon x\to y$ in $A$, and its morphisms from $f\colon x\to y$ to $g\colon z\to w$ are commutative squares

\begin{displaymath}
\itexarray{
  x & \to & z\\
  ^f \downarrow & & \downarrow^g\\
  y & \leftarrow & w }
\end{displaymath}
in $A$. There is a projection

\begin{displaymath}
r \colon Tw(A)^{op} \to A^{op}\times A
\end{displaymath}
sending $f\colon x\to y$ to $(y,x)$, and the coend of $H\colon A^{op}\times A\to C$ can be constructed as the colimit of $r^* H\colon Tw(A)^{op} \to C$.

Amazingly, this version needs no modification to become homotopical. Given $H\in D(A^{op}\times A)$ in a derivator, we can simply restrict along $r$ to $Tw(A)^{op}$, then take the (homotopy) colimit. To see that this agrees with the previous definition, it suffices to factor $q$ through $r$ via a [[homotopy final functor]] $s$:

\begin{displaymath}
\itexarray{ & & Tw(A)^{op} \\
  & ^{s}\nearrow & & \searrow^{r}\\
  (\Delta / A)^{op} & & \underset{q}{\to} & & A^{op}\times A }
\end{displaymath}
The definition of $s$ is simple: we regard an $n$-simplex as a string of $n$ composable arrows in $A$ and take its composite. The morphisms in the two categories match nicely. To show that $s$ is homotopy final, we must show that

\begin{displaymath}
\itexarray{(\Delta / A)^{op} & \xrightarrow{s} & Tw(A)^{op} \\
  \downarrow & & \downarrow \\
  * & \to & * }
\end{displaymath}
is a [[homotopy exact square]]. For this, it suffices to show that it becomes homotopy exact when pasted with any [[comma square]]

\begin{displaymath}
\itexarray{Q_{f} & \overset{}{\to} & *\\
  \downarrow & \swArrow & \downarrow^{f}\\
  (\Delta / A)^{op}& \underset{}{\to} & Tw(A)^{op}.}
\end{displaymath}
The objects of the category $Q_{f}$ are strings of composable $n\ge 2$ arrows whose composite is $f$. Its morphisms are like those of $(\Delta / A)^{op}$, but the first and last face maps are also given by composition instead of forgetting. In fact, it is precisely the category of simplices of the fiber of the [[two-sided bar construction]] $B(A(-,b),A,A(a,-))$ over $f$.

However, the simplicial map $B(A(-,b),A,A(a,-)) \to A(a,b)$, with $A(a,b)$ a discrete simplicial set, is well-known to be a [[simplicial homotopy equivalence]] and thus a weak equivalence of simplicial sets. Thus, each of its fibers is simplicially contractible, and hence each $Q_f$ has contractible nerve. This implies that

\begin{displaymath}
\itexarray{ Q_f & \to & * \\ \downarrow & & \downarrow \\ * & \to & * }
\end{displaymath}
is homotopy exact, and thus $s$ is homotopy final.

\hypertarget{as_a_bar_construction}{}\subsubsection*{{As a bar construction}}\label{as_a_bar_construction}

Another classical construction of a coend as a colimit is as the [[coequalizer]] of a [[parallel pair]]

\begin{displaymath}
\coprod_{f\colon x\to y} H(y,x) \;\rightrightarrows\; \coprod_{x} H(x,x).
\end{displaymath}
This can be obtained in a straightforward way from the previous construction. If $Ppr$ denotes the [[walking structure|walking]] parallel pair $(1 \rightrightarrows 0)$, then there is a functor $q\colon A^{tw}\to Ppr$ sending each object of $A$ to 0, each arrow of $A$ to 1, and sorting the morphisms by whether they map an arrow to its domain or to its codomain.

Then the above parallel pair is the (pointwise) [[left Kan extension]] of $p^* H \colon A^{tw} \to C$ along $q$. Because $q$ is a [[discrete opfibration]], left Kan extension along it can be computed with colimits over its fibers -- since each fiber is discrete, we obtain the coproducts above. And since the colimit of $p^*H$ is equivalently its left Kan extension to the [[point]], the functoriality of Kan extensions means that $\colim^{A^{tw}} p^* H$ is isomorphic to $\colim^{Ppr} Lan_q p^* H$, the latter being precisely the above coequalizer.

We can homotopify this in a straightforward way as well. Let $p\colon (\Delta / A)^{op} \to A^{op}\times A$ be as above, and let $q\colon (\Delta / A)^{op} \to \Delta^{op}$ be the obvious forgetful functor (whose target is the opposite of the [[simplex category]]). Note that as before, $Ppr$ is a final, but not homotopy-final, subcategory of $\Delta^{op}$. The functoriality of homotopy Kan extensions in a derivator means that the homotopy colimit of $p^* H\in D((\Delta / A)^{op})$ can equivalently be calculated as the homotopy colimit of $q_! p^* H \in D(\Delta^{op})$.

Note that since an object $D(\Delta^{op})$ is a [[simplicial object]] of $D$, it makes sense to call its colimit [[geometric realization]]. Moreover, the homotopy version of $q$ is also a discrete opfibration, and since pullbacks of fibrations are [[homotopy exact square|homotopy exact]], homotopy Kan extensions along $q$ are also computed as colimits over its fibers. These fibers are also discrete, so we obtain a simplicial diagram of the following sort:

\begin{displaymath}
\cdots \coprod_{x_0\xrightarrow{f_1} x_1 \xrightarrow{f_2} x_2} H(x_2,x_0)
\underoverset{\to}{\to}{\underoverset{\leftarrow}{\leftarrow}{\to}}
\coprod_{x_0\xrightarrow{f_1} x_1} H(x_1,x_0)
\underoverset{\to}{\to}{\leftarrow}
\coprod_{x} H(x,x)
\end{displaymath}
This is a derivator version of the [[bar construction]] of $H$. (A bar construction is perhaps the most classical construction of homotopy coends.)

\hypertarget{as_a_weighted_colimit}{}\subsubsection*{{As a weighted colimit}}\label{as_a_weighted_colimit}

A last classical construction of a coend is as a [[weighted colimit]] of $H\colon A^{op}\times A\to C$, weighted by the [[hom-functor]] $hom\colon A^{op}\times A \to Set$.

In order to homotopify this, recall that weighted colimits can be constructed in terms of Kan extensions by first Kan extending to the [[collage]] of the weighting (pro)functor, then restricting to the target object. Thus, let $B$ denote the collage of $hom\colon A^{op}\times A \to Set$ regarded as a profunctor $1 &#8696; A^{op}\times A$, with inclusions $u\colon 1\to B$ and $v\colon A^{op}\times A \to B$. We can therefore define the \textbf{homotopy coend} of $H\in D(A^{op}\times A)$ to be $u^* v_! H$.

To show that this is the same as the previous definitions, we simply observe that there is a comma square

\begin{displaymath}
\itexarray{Tw(A)^{op}& \overset{r}{\to} & A^{op}\times A\\
  ^!\downarrow & \swArrow_\alpha & \downarrow^v\\
  1& \underset{u}{\to} & B}
\end{displaymath}
Thus, by one of the axioms of a derivator, $u^* v_! H \cong \colim (r^* H)$.

[[!redirects end in a derivator]] [[!redirects ends in a derivator]] [[!redirects ends in derivators]] [[!redirects coend in a derivator]] [[!redirects coends in a derivator]] [[!redirects coends in derivators]]



\end{document}