FG \Rightarrow\mathrm{id}_{\sD}$ that satisfy the usual triangle

identities.

\item[-] A morphism of op\nbd{}prederivators $F : \sD\to\sD'$ is left

adjoint to $G : \sD' \to\sD$ (and $G$ is right adjoint to $F$) if there

exist $2$\nbd{}morphisms $\eta : \mathrm{id}_{\sD'}\Rightarrow GF$ and

$\epsilon : FG \Rightarrow\mathrm{id}_{\sD}$ that satisfy the usual

triangle identities.

\end{itemize}

\end{paragr}

The following three lemmas are easy $2$\nbd{}categorical routine and are left to

the reader.

The following three lemmas are easy $2$\nbd{}categorical routine and are left

to the reader.

\begin{lemma}\label{lemma:dereq}

Let $F : \sD\to\sD'$ be a morphism of op\nbd{}prederivators. If $F$ is an

equivalence then $\sD$ is a right op-derivator (resp.\ left op-derivator,

...

...

@@ -673,9 +681,9 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

quasi-inverse of $G$. Then, $F$ is left adjoint to $G$.

\end{lemma}

\begin{lemma}\label{lemma:ladjcocontinuous}

Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions and

$F : \sD\to\sD'$ a morphism of op\nbd{}prederivators. If $F$ is left adjoint

(of a morphism $G : \sD' \to\sD$), then it is cocontinuous.

Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions

and $F : \sD\to\sD'$ a morphism of op\nbd{}prederivators. If $F$ is left

adjoint (of a morphism $G : \sD' \to\sD$), then it is cocontinuous.

\end{lemma}

We end this section with a generalization of the notion of localization in the

context of op\nbd{}prederivators.

...

...

@@ -684,7 +692,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\[

\gamma_A : \C(A)\to\ho(\C(A))

\]

be the localization functor. This defines a strict morphism of

be the localization functor. The correspondence $A \mapsto\gamma_A$ is

natural in $A$ and defines a strict morphism of

op\nbd{}prederivators

\[

\gamma : \C\to\Ho(\C).

...

...

@@ -708,10 +717,10 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

such that for every small category $A$, $((\LL F)_A,\alpha_A)$ is the

\emph{absolute} total left derived functor of $F_A : \C(A)\to\C'(A)$. The

pair $(\LL F, \alpha)$ is unique up to a unique isomorphism and is referred

to as the \emph{left derived morphism of op\nbd{}prederivators of $F$}. Often, we

will discard $\alpha$ and simply refer to $\LL F$ as the left derived

morphism of $F$. The notion of \emph{strongly right derivable functor} is

defined dually and the notation $\mathbb{R}F$ is used.

to as the \emph{left derived morphism of op\nbd{}prederivators of $F$}.

Often, we will discard $\alpha$ and simply refer to $\LL F$ as the left

derived morphism of $F$. The notion of \emph{strongly right derivable

functor} is defined dually and the notation $\mathbb{R}F$ is used.

\end{definition}

\begin{example}

Let $(\C,\W)$ and $(\C,\W')$ be localizers and $F : \C\to\C'$ a functor.

...

...

@@ -738,11 +747,11 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\]

\end{proposition}

\begin{proof}

Let $\alpha : F' \circ\gamma\Rightarrow F\circ\gamma$ be the$2$\nbd{}morphism

of op\nbd{}prederivators defined \emph{mutatis mutandis} as in

\ref{paragr:prelimgonzalez} but at the level of op\nbd{}prederivators.

Let $\alpha : F' \circ\gamma\Rightarrow F\circ\gamma$ be the

$2$\nbd{}morphism of op\nbd{}prederivators defined \emph{mutatis mutandis}

as in \ref{paragr:prelimgonzalez} but at the level of op\nbd{}prederivators.

Proposition \ref{prop:gonz} gives us that for every small category $A$, the

functor $F'_A$ is absolutely totally left derivable with $(F'_A,\alpha_A)$

functor $F_A$ is absolutely totally left derivable with $(F'_A,\alpha_A)$

its total left derived functor. This means exactly that $F'$ is strongly

left derivable and $(F',\alpha)$ is the left derived morphism of

op\nbd{}prederivators of $F$.

...

...

@@ -750,8 +759,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\section{Homotopy cocartesian squares}

\begin{paragr}

Let $\Delta_1$ be the ordered set $\{0 <1\}$ seen as category. We use the

notation $\square$ for the category $\Delta_1\times\Delta_1$, which can be pictured

as the \emph{commutative} square

notation $\square$ for the category $\Delta_1\times\Delta_1$, which can be

pictured as the \emph{commutative} square

\[

\begin{tikzcd}

(0,0)\ar[r]\ar[d]&(0,1)\ar[d]\\

...

...

@@ -847,9 +856,9 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\]

\end{paragr}

\begin{proposition}

Let $\sD$ be a right op\nbd{}prederivator. An object $X$ of $\sD(\square)$ is

cocartesian if and only if the canonical map $ p_!(i^*_{\ulcorner}(X))\to

X_{(1,1)}$ is an isomorphism.

Let $\sD$ be a right op\nbd{}prederivator. An object $X$ of $\sD(\square)$

is cocartesian if and only if the canonical map $ p_!(i^*_{\ulcorner}(X))

\toX_{(1,1)}$ is an isomorphism.

\end{proposition}

\begin{proof}

Let $Y$ be another object of $\sD(\square)$. Using the adjunction

...

...

@@ -865,7 +874,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

!}(i_{\ulcorner}^*(X))$ is an isomorphism. The rest follows then from

\cite[Lemma 9.2.2(i)]{groth2013book}.

\end{proof}

Hence, for a homotopy cocomplete localizer $(\C,\W)$, a square is homotopy

Hence, for a homotopy cocomplete localizer $(\C,\W)$, a commutative square of

$\C$ is homotopy

cocartesian if and only if the bottom right apex of the square is the homotopy

colimit of upper left corner of the square. This hopefully justify the

terminology of ``cocartesian square''.

...

...

@@ -919,10 +929,10 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

3.13(1)]{groth2013derivators}.

\end{proof}

\section{Model categories}

In this section, we quickly review some aspects of the relation between Quillen's

theory of model categories and Grothendieck's theory of derivators. We suppose

that the reader is familiar with the former one and refer to the standard

textbooks on the subject (such as \cite{hovey2007model},

In this section, we quickly review some aspects of the relation between

Quillen's theory of model categories and Grothendieck's theory of derivators.

We suppose that the reader is familiar with the former one and refer to the

standard textbooks on the subject (such as \cite{hovey2007model},

\cite{hirschhorn2009model} or \cite{dwyer1995homotopy}) for basic definitions

and results.

...

...

@@ -944,8 +954,9 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\begin{theorem}[Cisinski]\label{thm:cisinskiII}

Let $\M$ and $\M'$ be two model categories and ${F : \M\to\M'}$ a left

Quillen functor (i.e.\ the left adjoint in a Quillen adjunction). The

functor $F$ is strongly left derivable and the morphism of op\nbd{}prederivators

$\LL F : \Ho(\M)\to\Ho(\M')$ is homotopy cocontinuous.

functor $F$ is strongly left derivable and the morphism of

op\nbd{}prederivators $\LL F : \Ho(\M)\to\Ho(\M')$ is homotopy

cocontinuous.

\end{theorem}

\begin{remark}

The obvious duals of the two above theorems are also true. The reason we put

...

...

@@ -954,28 +965,29 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\end{remark}

\begin{remark}

Since the homotopy op\nbd{}prederivator of a model category

$(\M,\W,\Cof,\Fib)$ only depends on its underlying localizer, the

existence of the classes $\Cof$ and $\Fib$ with the usual

properties defining model structure ought to be thought as a

\emph{property} of the localizer $(\M,\W)$, which is sufficcient

to define a ``homotopy theory''. For example, Theorem

\ref{thm:cisinskiI} should have been stated by saying that if a

localizer $(\M,\W)$ can be extended to a model category

$(\M,\W,\Cof,\Fib)$ only depends on its underlying localizer, the existence

of the classes $\Cof$ and $\Fib$ with the usual properties defining model

structure ought to be thought as a \emph{property} of the localizer

$(\M,\W)$, which is sufficcient to define a ``homotopy theory''. For

example, Theorem \ref{thm:cisinskiI} should have been stated by saying that

if a localizer $(\M,\W)$ can be extended to a model category

$(\M,\W,\Cof,\Fib)$, then it is homotopy cocomplete.

\end{remark}

Even if Theorem \ref{thm:cisinskiI} tells us that (the homotopy

op\nbd{}prederivator of) a model category $(\M,\W,\Cof,\Fib)$ have homotopy left

Kan extensions, it is not generally true that for a small category $A$ the

category of diagrams $\M(A)$ admits a model structure with the pointwise weak equivalences as its weak

equivalences. Hence, in general we cannot use the theory of Quillen functors to compute

homotopy left Kan extensions (and in particular homotopy colimits). However,

in practice all the model categories that we shall encounter are

\emph{cofibrantly generated}, in which case the theory is much simpler because $\M(A)$ does admit a model structure with the pointwise weak equivalences as its weak equivalences.

op\nbd{}prederivator of) a model category $(\M,\W,\Cof,\Fib)$ have homotopy

left Kan extensions, it is not generally true that for a small category $A$

the category of diagrams $\M(A)$ admits a model structure with the pointwise

weak equivalences as its weak equivalences. Hence, in general we cannot use

the theory of Quillen functors to compute homotopy left Kan extensions (and in

particular homotopy colimits). However, in practice all the model categories

that we shall encounter are \emph{cofibrantly generated}, in which case the

theory is much simpler because $\M(A)$ does admit a model structure with the

pointwise weak equivalences as its weak equivalences.

\begin{paragr}\label{paragr:cofprojms}

Let $\C$ be a category with coproducts and $A$ a small category. For every

object $X$ of $\C$ and every object $a$ of $A$, we define $X\otimes a$ as the

functor

object $X$ of $\C$ and every object $a$ of $A$, we define $X\otimes a$ as

the functor

\[

\begin{aligned}

X \otimes a : A &\to\C\\

...

...

@@ -994,12 +1006,12 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\begin{proposition}\label{prop:modprs}

Let $(\M,\W,\Cof,\Fib)$ be a cofibrantly generated model category with $I$

(resp.\ $J$) as a set of generating cofibrations (resp. trivial

cofibrations). For every small category $A$, there exists a model structure on

$\M(A)$ such that:

cofibrations). For every small category $A$, there exists a model structure

on $\M(A)$ such that:

\begin{itemize}[label=-]

\item weak equivalences are pointwise weak equivalences,

\item fibrations are pointwise fibrations,

\item cofibrations are those morphisms which have the left lifting property

\itemthe weak equivalences are the pointwise weak equivalences,

\itemthe fibrations are the pointwise fibrations,

\itemthe cofibrations are those morphisms which have the left lifting property

to trivial fibrations. \end{itemize} Moreover, this model structure is

cofibrantly generated and a set of generating cofibration (resp.\ trivial

cofibrations) is given by

...

...

@@ -1022,8 +1034,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.

\emph{projective model structure on $\M(A)$}.

\end{paragr}

\begin{proposition}

Let $(\M,\W,\Cof,\Fib)$ be a cofibrantely generated model

category. For every $u : A \to B$, the adjunction

Let $(\M,\W,\Cof,\Fib)$ be a cofibrantely generated model category. For