Commit c4d240c8 authored by Leonard Guetta's avatar Leonard Guetta
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still editing typos

parent 0df23338
......@@ -238,7 +238,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
$\oo$\nbd{}categories $\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant
and \good{}, it follows from Corollary \ref{cor:usefulcriterion} and
an immediate induction that all $\sk_k(C)$ are \good{}. The result
follows then from Lemma \ref{lemma:filtration} and Proposition
follows then from Lemma \ref{lemma:filtration}, Corollary \ref{cor:cofprojms} and Proposition
\ref{prop:sequentialhmtpycolimit}.
\end{proof}
\section{The miraculous case of 1-categories}
......
......@@ -74,9 +74,9 @@ For later reference, we put here the following definition.
\end{definition}
\begin{paragr}
A \emph{morphism of localizers} $F : (\C,\W) \to (\C',\W')$ is a functor
$F:\C\to\C'$ that preserves weak equivalences, i.e.\ such that $F(\W) \subseteq
\W'$. The universal property of the localization implies that $F$ induces a
canonical functor
$F:\C\to\C'$ that preserves weak equivalences, i.e.\ such that $F(\W)
\subseteq \W'$. The universal property of the localization implies that $F$
induces a canonical functor
\[
\overline{F} : \ho(\C) \to \ho(\C')
\]
......@@ -88,8 +88,8 @@ For later reference, we put here the following definition.
\end{tikzcd}
\]
is commutative. Let $G : (\C,\W) \to (\C',\W')$ be another morphism of
localizers. A \emph{$2$\nbd{}morphism of localizers} from $F$ to $G$ is simply a
natural transformation $\alpha : F \Rightarrow G$. The universal property of
localizers. A \emph{$2$\nbd{}morphism of localizers} from $F$ to $G$ is simply
a natural transformation $\alpha : F \Rightarrow G$. The universal property of
the localization implies that there exists a unique natural transformation
\[
\begin{tikzcd} \ho(\C) \ar[r,bend left,"\overline{F}",""{name=A,below}]
......@@ -108,10 +108,10 @@ For later reference, we put here the following definition.
is commutative in an obvious sense.
\end{paragr}
\begin{remark}\label{remark:localizedfunctorobjects}
Since we always consider that for every localizer $(\C,\W)$ the categories $\C$
and $\ho(\C)$ have the same class of objects and the localization functor is the
identity on objects, it follows that for a morphism of localizers ${F : (\C,\W)
\to (\C',\W')}$, we tautologically have
Since we always consider that for every localizer $(\C,\W)$ the categories
$\C$ and $\ho(\C)$ have the same class of objects and the localization functor
is the identity on objects, it follows that for a morphism of localizers ${F :
(\C,\W) \to (\C',\W')}$, we tautologically have
\[
\overline{F}(X)=F(X)
\]
......@@ -143,8 +143,8 @@ For later reference, we put here the following definition.
$F$}. Often we will abusively discard $\alpha$ and simply refer to $\LL F$
as the total left derived functor of $F$.
The notion of \emph{total right derivable functor} is defined dually and denoted by
$\RR F$ when it exists.
The notion of \emph{total right derivable functor} is defined dually and
denoted by $\RR F$ when it exists.
\end{paragr}
\begin{example}\label{rem:homotopicalisder}
Let $(\C,\W)$ and $(\C',\W')$ be two localizers and $F: \C \to \C'$ be a
......@@ -161,7 +161,7 @@ we shall use in the sequel.
whose unit is denoted by $\eta$. Suppose that $G$ is totally right derivable
with $(\RR G,\beta)$ its total right derived functor and suppose that $\RR G$
has a left adjoint $F' : \ho(\C) \to \ho(\C')$; the co-unit of this last
adjunction being denoted by $\epsilon'$. All this data induce a natural
adjunction being denoted by $\epsilon'$. All this data induces a natural
transformation $\alpha : F' \circ \gamma \Rightarrow \gamma' \circ F$ defined
as the following composition
\[
......@@ -192,14 +192,15 @@ we shall use in the sequel.
% faudrait vérifier que je n'ai pas dit de bêtises en le reformulant.}
\section{(op-)Derivators and homotopy colimits}
\begin{notation}We denote by $\CCat$ the $2$\nbd{}category of small categories and
$\CCAT$ the $2$\nbd{}category of big categories. For a $2$\nbd{}category
$\underline{A}$, the $2$\nbd{}category obtained from $\underline{A}$ by switching
the source and targets of $1$-cells is denoted by $\underline{A}^{\op}$.
\begin{notation}We denote by $\CCat$ the $2$\nbd{}category of small categories
and $\CCAT$ the $2$\nbd{}category of big categories. For a $2$\nbd{}category
$\underline{A}$, the $2$\nbd{}category obtained from $\underline{A}$ by
switching the source and targets of $1$-cells is denoted by
$\underline{A}^{\op}$.
The terminal category, i.e.\ the category with only one object and no
non-trivial arrows, is canonically denoted by $e$. For a (small) category $A$,
the canonical morphism from $A$ to $e$ is denoted by
the unique functor from $A$ to $e$ is denoted by
\[
p_A : A \to e.
\]
......@@ -229,7 +230,7 @@ we shall use in the sequel.
\]
with $A$ and $B$ small categories,
\end{itemize}
compatible with compositions and units in a strict sense. \iffalse such that
compatible with compositions and units (in a strict sense).\iffalse such that
the following axioms are satisfied:
\begin{itemize}[label=-]
\item for every small category $A$, $(1_A)^*=1_{\sD(A)}$,
......@@ -260,22 +261,24 @@ we shall use in the sequel.
\end{definition}
\begin{remark}
Note that some authors call \emph{prederivator} what we have called
\emph{op\nbd{}prederivator}. The terminology we chose in the above definition is
compatible with the original one of Grothendieck, who called
\emph{op\nbd{}prederivator}. The terminology we chose in the above definition
is compatible with the original one of Grothendieck, who called
\emph{prederivator} a $2$\nbd{}functor from $\CCat$ to $\CCAT$ that is
contravariant at the level of $1$-cells \emph{and} at the level of $2$\nbd{}cells.
contravariant at the level of $1$-cells \emph{and} at the level of
$2$\nbd{}cells.
\end{remark}
\begin{example}\label{ex:repder}
Let $\C$ be a category. For a small category $A$, we use the notation $\C(A)$
for the category $\underline{\Hom}(A,\C)$ of functors $A \to \C$ and natural
transformations between them. The correspondence $A \mapsto \C(A)$ is
$2$\nbd{}functorial in an obvious sense and thus defines an op\nbd{}prederivator
$2$\nbd{}functorial in an obvious sense and thus defines an
op\nbd{}prederivator
\begin{align*}
\C : \CCat^{\op} &\to \CCAT \\
A &\mapsto \C(A)
\end{align*}
which we call the op\nbd{}prederivator \emph{represented by $\C$}. For $u : A \to
B$ in $\CCat$,
which we call the op\nbd{}prederivator \emph{represented by $\C$}. For $u : A
\to B$ in $\CCat$,
\[
u^* : \C(A) \to \C(B)
\]
......@@ -284,11 +287,13 @@ we shall use in the sequel.
We now turn to the most important way of obtaining op\nbd{}prederivators.
\begin{paragr}\label{paragr:homder}
Let $(\C,\W)$ be a localizer. For every small category $A$, we write $\W_A$ the class of \emph{pointwise weak
equivalences} of the category $\C(A)$, i.e.\ the class of arrows $\alpha : d \to d'$ of $\C(A)$ such that
$\alpha_a : d(a) \to d'(a)$ belongs to $\W$ for every $a \in \Ob(A)$. This defines a localizer $(\C(A),\W_A)$.
The correspondence $A \mapsto (\C(A),\W_A)$ is $2$\nbd{}functorial in that every $u
: A \to B$ induces by pre-composition a morphism of localizers
Let $(\C,\W)$ be a localizer. For every small category $A$, we write $\W_A$
the class of \emph{pointwise weak equivalences} of the category $\C(A)$, i.e.\
the class of arrows $\alpha : d \to d'$ of $\C(A)$ such that $\alpha_a : d(a)
\to d'(a)$ belongs to $\W$ for every $a \in \Ob(A)$. This defines a localizer
$(\C(A),\W_A)$. The correspondence $A \mapsto (\C(A),\W_A)$ is
$2$\nbd{}functorial in that every $u : A \to B$ induces by pre-composition a
morphism of localizers
\[
u^* : (\C(B),\W_B) \to (\C(A),\W_A)
\]
......@@ -303,14 +308,15 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\ar[from=A,to=B,Rightarrow,"\alpha^*"]
\end{tikzcd}
\]
(This last property is trivial since a $2$\nbd{}morphism of localizers is simply a
natural transformation between the underlying functors.) Then, by the
universal property of the localization, every morphism $u : A \to B$ of $\Cat$ induces a functor, again denoted by $u^*$,
(This last property is trivial since a $2$\nbd{}morphism of localizers is
simply a natural transformation between the underlying functors.) Then, by the
universal property of the localization, every morphism $u : A \to B$ of $\Cat$
induces a functor, again denoted by $u^*$,
\[
u^* : \ho(\C(B)) \to \ho(\C(A))
\]
and every natural transformation $\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}]
\ar[r,bend right, "v"',""{name=B,above}] & B
and every natural transformation $\begin{tikzcd}A \ar[r,bend
left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B
\ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ induces natural
transformation, again denoted by $\alpha^*$,
\[
......@@ -325,29 +331,30 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\Ho^{\W}(\C) : \CCat^{\op} &\to \CCAT\\
A &\mapsto \ho(\C(A)),
\end{align*}
which we call the \emph{homotopy op\nbd{}prederivator of $(\C,\W)$}. When there is
no risk of confusion we will simply write $\Ho(\C)$ instead of $\Ho^{\W}(\C)$.
All the op\nbd{}prederivators we shall work with arise this way. Notice that for
the terminal category $e$, we have a canonical isomorphism
which we call the \emph{homotopy op\nbd{}prederivator of $(\C,\W)$}. When
there is no risk of confusion we will simply write $\Ho(\C)$ instead of
$\Ho^{\W}(\C)$. All the op\nbd{}prederivators we shall work with arise this
way. Notice that for the terminal category $e$, we have a canonical
isomorphism
\[
\Ho(\C)(e)\simeq \ho(\C),
\]
which we shall use without further reference.
\end{paragr}
\begin{definition}
An op\nbd{}prederivator $\sD$ has \emph{left Kan extensions} if for every $u : A
\to B$ in $\Cat$, the functor $ u^* : \sD(B) \to \sD(A)$ has a left adjoint
An op\nbd{}prederivator $\sD$ has \emph{left Kan extensions} if for every $u :
A \to B$ in $\Cat$, the functor $ u^* : \sD(B) \to \sD(A)$ has a left adjoint
\[
u_! : \sD(A) \to \sD(B).
\]
\end{definition}
\begin{example}
Let $\C$ be a category. The op\nbd{}prederivator represented by $\C$ has left Kan
extensions if and only if the category $\C$ has left Kan extensions along every
morphism $u : A \to B$ of $\Cat$ in the usual sense. By a standard categorical
argument, this means that the op\nbd{}prederivator represented by $\C$ has left Kan
extensions if and only if $\C$ is cocomplete. Note that for every small category
$A$, the functor
Let $\C$ be a category. The op\nbd{}prederivator represented by $\C$ has left
Kan extensions if and only if the category $\C$ has left Kan extensions along
every morphism $u : A \to B$ of $\Cat$ in the usual sense. By a standard
categorical argument, this means that the op\nbd{}prederivator represented by
$\C$ has left Kan extensions if and only if $\C$ is cocomplete. Note that for
every small category $A$, the functor
\[
p_A^* : \C \simeq \C(e) \to \C(A)
\]
......@@ -359,10 +366,10 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\]
\end{example}
\begin{paragr}
We say that a localizer $(\C,\W)$ has \emph{homotopy left Kan extensions} when the
homotopy op\nbd{}prederivator of $(\C,\W)$ has left Kan extensions. In this case,
for every small category $A$, the \emph{homotopy colimit functor of $A$-shaped
diagrams} is defined as
We say that a localizer $(\C,\W)$ has \emph{homotopy left Kan extensions} when
the homotopy op\nbd{}prederivator of $(\C,\W)$ has left Kan extensions. In
this case, for every small category $A$, the \emph{homotopy colimit functor of
$A$-shaped diagrams} is defined as
\[
\hocolim_A := p_{A!} : \ho(\C(A)) \to \ho(\C).
\]
......@@ -399,8 +406,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
C \ar[r,"g"'] & D \ar[from=1-2,to=2-1,Rightarrow,"\alpha"]
\end{tikzcd}
\]
be a $2$\nbd{}square in $\CCat$. Every op\nbd{}prederivator $\sD$
induces a $2$\nbd{}square:
be a $2$\nbd{}square in $\CCat$. Every op\nbd{}prederivator $\sD$ induces a
$2$\nbd{}square:
\[
\begin{tikzcd}
\sD(A) & \sD(B) \ar[l,"f^*"'] \\
......@@ -462,8 +469,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
A \emph{right op-derivator} is an op\nbd{}prederivator $\sD$ such that the
following axioms are satisfied:
\begin{description}
\item[Der 1)] For every finite family $(A_i)_{i \in I}$ of small categories, the
canonical functor
\item[Der 1)] For every finite family $(A_i)_{i \in I}$ of small categories,
the canonical functor
\[
\sD(\amalg_{i \in I}A_i) \to \prod_{i \in I}\sD(A_i)
\]
......@@ -510,29 +517,29 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{example}
\begin{remark}
Beware not to generalize the previous example too hastily. It is not true in
general that axiom \textbf{Der 3d} implies axiom \textbf{Der 4d}; even in the case
of the homotopy op\nbd{}prederivator of a localizer.
general that axiom \textbf{Der 3d} implies axiom \textbf{Der 4d}; even in
the case of the homotopy op\nbd{}prederivator of a localizer.
\end{remark}
This motivates the following definition.
\begin{definition}\label{def:cocompletelocalizer}
A localizer $(\C,\W)$ is \emph{homotopy cocomplete} if the op\nbd{}prederivator
$\Ho(\C)$ is a right op-derivator.
A localizer $(\C,\W)$ is \emph{homotopy cocomplete} if the
op\nbd{}prederivator $\Ho(\C)$ is a right op-derivator.
\end{definition}
\begin{paragr}
Axioms \textbf{Der 3d} and \textbf{Der 4d} can be dualized to obtain axioms
\textbf{Der 3g} and \textbf{Der 4g}, which informally say that the
op\nbd{}prederivator has right Kan extensions and that they are computed
pointwise. An op\nbd{}prederivator satisfying axioms \textbf{Der 1}, \textbf{Der
2}, \textbf{Der 3g} and \textbf{Der 4g} is a \emph{left op-derivator}. In
fact, an op\nbd{}prederivator $\sD$ is a left op-derivator if and only if the
op\nbd{}prederivator
pointwise. An op\nbd{}prederivator satisfying axioms \textbf{Der 1},
\textbf{Der 2}, \textbf{Der 3g} and \textbf{Der 4g} is a \emph{left
op-derivator}. In fact, an op\nbd{}prederivator $\sD$ is a left
op-derivator if and only if the op\nbd{}prederivator
\begin{align*}
\CCat &\to \CCAT \\
A &\mapsto (\sD(A^{\op}))^{\op}
\end{align*}
is a right op\nbd{}prederivator. An op\nbd{}prederivator which is both a left and
right op-derivator is an \emph{op-derivator}. For details, the reader can
refer to any of the references on derivators previously cited.
is a right op\nbd{}prederivator. An op\nbd{}prederivator which is both a
left and right op-derivator is an \emph{op-derivator}. For details, the
reader can refer to any of the references on derivators previously cited.
\end{paragr}
\section{Morphisms of op-derivators, preservation of homotopy colimits}
We refer to \cite{leinster1998basic} for the precise definitions of
......@@ -540,8 +547,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
modification.
\begin{paragr}
Let $\sD$ and $\sD'$ be two op\nbd{}prederivators. A \emph{morphism of
op\nbd{}prederivators} $F : \sD \to \sD'$ is a pseudo-natural transformation
from $\sD$ to $\sD'$. This means that $F$ consists of:
op\nbd{}prederivators} $F : \sD \to \sD'$ is a pseudo-natural
transformation from $\sD$ to $\sD'$. This means that $F$ consists of:
\begin{itemize}[label=-]
\item a functor $F_A : \sD(A) \to \sD'(A)$ for every small category $A$,
\item an isomorphism of functors $F_u: F_A u^* \overset{\sim}{\Rightarrow}
......@@ -559,22 +566,23 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\emph{strict} when $F_u$ is an identity for every $u : A \to B$.
Let $F : \sD \to \sD'$ and $G : \sD \to \sD'$ be morphisms of
op\nbd{}prederivators. A \emph{$2$\nbd{}morphism $\phi : F \Rightarrow G$} is a modification
from $F$ to $G$. This means that $F$ consists of a natural transformation
$\phi_A : F_A \Rightarrow G_A$ for every small category $A$, and is subject
to a coherence axiom similar to the one for natural transformations.
op\nbd{}prederivators. A \emph{$2$\nbd{}morphism $\phi : F \Rightarrow G$}
is a modification from $F$ to $G$. This means that $F$ consists of a natural
transformation $\phi_A : F_A \Rightarrow G_A$ for every small category $A$,
and is subject to a coherence axiom similar to the one for natural
transformations.
We denote by $\PPder$ the $2$\nbd{}category of op\nbd{}prederivators, morphisms of
op\nbd{}prederivators and $2$\nbd{}morphisms of op\nbd{}prederivators.
We denote by $\PPder$ the $2$\nbd{}category of op\nbd{}prederivators,
morphisms of op\nbd{}prederivators and $2$\nbd{}morphisms of
op\nbd{}prederivators.
\end{paragr}
\begin{example}
Let $F : \C \to \D$ be a functor. It induces a strict morphism at
the level of op\nbd{}prederivators, again denoted by $F$, where
for every small category $A$, the functor $F_A : \C(A) \to \C'(A)$
is induced by post-composition. Similarly, every natural
transformation induces a $2$\nbd{}morphism at the level of
represented op\nbd{}prederivators.
Let $F : \C \to \D$ be a functor. It induces a strict morphism at the level
of op\nbd{}prederivators, again denoted by $F$, where for every small
category $A$, the functor $F_A : \C(A) \to \C'(A)$ is induced by
post-composition. Similarly, every natural transformation induces a
$2$\nbd{}morphism at the level of represented op\nbd{}prederivators.
\end{example}
\begin{example}
Let $F : (\C,\W) \to (\C',\W')$ be a morphism of localizers. For every small
......@@ -601,11 +609,11 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
where $\Loc$ is the $2$\nbd{}category of localizers.
\end{example}
\begin{paragr}\label{paragr:canmorphismcolimit}
Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions and
let $F : \sD \to \sD'$ morphism of op\nbd{}prederivators. For every $u : A \to
B$, there is a canonical natural transformation
Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions
and let $F : \sD \to \sD'$ morphism of op\nbd{}prederivators. For every $u :
A \to B$, there is a canonical natural transformation
\[
u_! \circ F_A \Rightarrow F_B \circ u_!
u_!\, F_A \Rightarrow F_B\, u_!
\]
defined as
\[
......@@ -617,52 +625,52 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\ar[from=2-3,to=B,Rightarrow,"\epsilon"]
\end{tikzcd}
\]
For example, when $\sD$ is the homotopy op\nbd{}prederivator of a
localizer and $B$ is the terminal category $e$, for every $X$ object
of $\sD(A)$ the previous canonical morphism reads
For example, when $\sD$ is the homotopy op\nbd{}prederivator of a localizer
and $B$ is the terminal category $e$, for every $X$ object of $\sD(A)$ the
previous canonical morphism reads
\[
\hocolim_{A}(F_A(X))\to F_e(\hocolim_A(X)).
\]
\end{paragr}
\begin{definition}\label{def:cocontinuous}
Let $F : \sD \to \sD'$ morphism of op\nbd{}prederivators and suppose that $\sD$
and $\sD'$ both admit left Kan extensions. We say that $F$ is
Let $F : \sD \to \sD'$ morphism of op\nbd{}prederivators and suppose that
$\sD$ and $\sD'$ both admit left Kan extensions. We say that $F$ is
\emph{cocontinuous} or \emph{left exact} if for every $u: A \to B$, the
canonical morphism
\[
u_! \circ F_A \Rightarrow F_B \circ u_!
u_! \, F_A \Rightarrow F_B \, u_!
\]
is an isomorphism.
\end{definition}
\begin{remark}
When $\sD$ and $\sD'$ are homotopy op\nbd{}prederivators we will often say that a
morphism $F : \sD \to \sD'$ is \emph{homotopy cocontinuous} instead of
\emph{cocontinuous} to emphasize the fact that it preserves homotopy Kan
When $\sD$ and $\sD'$ are homotopy op\nbd{}prederivators we will often say
that a morphism $F : \sD \to \sD'$ is \emph{homotopy cocontinuous} instead
of \emph{cocontinuous} to emphasize the fact that it preserves homotopy Kan
extensions.
\end{remark}
\begin{example}
Let $F : \C \to \C'$ be a functor and suppose that $\C$ and $\C'$ are
cocomplete. The morphism induced by $F$ at the level of represented
op\nbd{}prederivators is cocontinuous if and only if $F$ is cocontinuous in the
usual sense.
op\nbd{}prederivators is cocontinuous if and only if $F$ is cocontinuous in
the usual sense.
\end{example}
\begin{paragr}\label{paragr:prederequivadjun}
As in any $2$\nbd{}category, the notions of equivalence and adjunction make sense
in $\PPder$. Precisely, we have that:
As in any $2$\nbd{}category, the notions of equivalence and adjunction make
sense in $\PPder$. Precisely, we have that:
\begin{itemize}
\item[-] A morphism of op\nbd{}prederivators $F : \sD \to \sD'$ is an equivalence
when there exists a morphism $G : \sD' \to \sD$ such that $FG$ is
isomorphic to $\mathrm{id}_{\sD'}$ and $GF$ is isomorphic to
\item[-] A morphism of op\nbd{}prederivators $F : \sD \to \sD'$ is an
equivalence when there exists a morphism $G : \sD' \to \sD$ such that $FG$
is isomorphic to $\mathrm{id}_{\sD'}$ and $GF$ is isomorphic to
$\mathrm{id}_{\sD}$; the morphism $G$ is a \emph{quasi-inverse} of $F$.
\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 $F$) when there exist
$2$\nbd{}morphisms $\eta : \mathrm{id}_{\sD'} \Rightarrow GF$ and $\epsilon :
FG \Rightarrow \mathrm{id}_{\sD}$ 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))
\to X_{(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
\item the weak equivalences are the pointwise weak equivalences,
\item the fibrations are the pointwise fibrations,
\item the 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
every $u : A \to B$, the adjunction
\[
\begin{tikzcd}
u_! : \M(A) \ar[r,shift left]& \ar[l,shift left] \M(B) : u^*
......@@ -1037,7 +1049,8 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.