\chapter{Homotopical algebra} The present chapter stands out from the others as it contains no original results. Its goal is simply to introduce the language and tools of homotopical algebra that we shall need in the rest of the dissertation. Consequently, most of the results are simply asserted and the reader will find references to the literature for the proofs. The main notion of homotopical algebra we aim for is the one of \emph{homotopy colimits} and our language of choice is that of Grothendieck's theory of \emph{derivators} \cite{grothendieckderivators}. We do not assume that the reader is familiar with this theory and will quickly recall the basics. If needed, gentle introductions can be found in \cite{maltsiniotis2001introduction} and in a letter from Grothendieck to Thomason \cite{grothendieck1991letter}; more detailed introductions can be found in \cite{groth2013derivators} and in the first section of \cite{cisinski2003images}; finally, a rather complete (yet unfinished and unpublished) textbook on the subject is \cite{groth2013book}. \iffalse Let us quickly motive this choice for the reader unfamiliar with this theory. From an elementary point of view, a homotopy theory is given (or rather \emph{presented by}) by a category $\C$ and a class $\W$ of arrows of $\C$, which we traditionally refer to as \emph{weak equivalences}. The point of homotopy theory is to consider that the objects of $\C$ connected by a zigzag of weak equivalences should be indistinguishable. From a category theorist perspective, a most natural One of the most basic invariant associated to such a data is the localisation of $\C$ with respect to $\W$. That is to say, the category $\ho^{\W}(\C)$ obtained from $\C$ by forcing the arrows of $\W$ to become isomorphisms. While the problem'' is that the category $\ho^{\W}(\C)$ is poorly behaved. For example, \fi \section{Localization, derivation} \begin{paragr}\label{paragr:loc} A \emph{localizer} is a pair $(\C,\W)$ where $\C$ is a category and $\W$ is a class of arrows of $\C$, which we usually refer to as the \emph{weak equivalences}. We denote by $\ho^{\W}(\C)$, or simply $\ho(\C)$ when there is no ambiguity, the localization of $\C$ with respect to $\W$ and by $\gamma : \C \to \ho(\C)$ the localization functor \cite[1.1]{gabriel1967calculus}. Recall the universal property of the localization: for every category $\D$, the functor induced by pre-composition $\gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D)$ is fully faithful and its essential image consists of functors $F~:~\C~\to~\D$ that send the morphisms of $\W$ to isomorphisms of $\D$. We shall always consider that $\C$ and $\ho(\C)$ have the same class of objects and implicitly use the equality $\gamma(X)=X$ for every object $X$ of $\C$. The class of arrows $\W$ is said to be \emph{saturated} when we have the property: $f \in \W \text{ if and only if } \gamma(f) \text{ is an isomorphism. }$ \end{paragr} For later reference, we put here the following definition. \begin{definition}\label{def:couniversalwe} Let $(\C,\W)$ be a localizer such that $\C$ has amalgamated sums. A morphism $f : X \to Y$ in $\W$ is a \emph{co-universal weak equivalence} if for every cocartesian square of the form $\begin{tikzcd} X \ar[r] \ar[d,"f"] & X' \ar[d,"f'"] \\ Y \ar[r] & Y', \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd}$ the morphism $f'$ is also a weak equivalence. \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 $\overline{F} : \ho(\C) \to \ho(\C')$ such that the square $\begin{tikzcd} \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\ \ho(\C) \ar[r,"\overline{F}"] & \ho(\C'). \end{tikzcd}$ 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 the localization implies that there exists a unique natural transformation $\begin{tikzcd} \ho(\C) \ar[r,bend left,"\overline{F}",""{name=A,below}] \ar[r,bend right,"\overline{G}"',""{name=B,above}] & \ho(\C') \ar[from=A,to=B,Rightarrow,"\overline{\alpha}"]\end{tikzcd}$ such that the $2$\nbd{}diagram $\begin{tikzcd}[row sep=huge] \C\ar[d,"\gamma"] \ar[r,bend left,"F",""{name=A,below}] \ar[r,bend right,"G"',""{name=B,above}] & \C'\ar[d,"\gamma'"] \ar[from=A,to=B,Rightarrow,"\alpha"] \\ \ho(\C) \ar[r,bend left,"\overline{F}",""{name=A,below}] \ar[r,bend right,"\overline{G}"',""{name=B,above}] & \ho(\C') \ar[from=A,to=B,Rightarrow,"\overline{\alpha}"] \end{tikzcd}$ 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 $F \colon (\C,\W) \to (\C',\W')$, we tautologically have $\overline{F}(X)=F(X)$ for every object $X$ of $\C$. \end{remark} \begin{paragr}\label{paragr:defleftderived} Let $(\C,\W)$ and $(\C',\W')$ be two localizers. A functor $F : \C \to \C'$ is \emph{totally left derivable} when there exists a functor $\LL F : \ho(\C) \to \ho(\C')$ and a natural transformation $\alpha : \LL F \circ \gamma \Rightarrow \gamma'\circ F$ that makes $\LL F$ the \emph{right} Kan extension of $\gamma' \circ F$ along $\gamma$: $\begin{tikzcd} \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\ \ho(\C) \ar[r,"\LL F"'] & \ho(\C'). \arrow[from=2-1, to=1-2,"\alpha",Rightarrow] \end{tikzcd}$ When this right Kan extension is \emph{absolute}, we say that $F$ is \emph{absolutely totally left derivable}. When a functor $F$ is totally left derivable, the pair $(\LL F,\alpha)$ is unique up to a unique natural isomorphism and is referred to as \emph{the total left derived functor of $F$}. 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. \end{paragr} \begin{example}\label{rem:homotopicalisder} Let $(\C,\W)$ and $(\C',\W')$ be two localizers and $F: \C \to \C'$ be a functor. If $F$ preserves weak equivalences (i.e.\ it is a morphism of localizers) then the universal property of localization implies that $F$ is absolutely totally left and right derivable and $\LL F \simeq \RR F \simeq \overline{F}$. \end{example} To end this section, we recall a derivability criterion due to Gonzalez, which we shall use in the sequel. \begin{paragr}\label{paragr:prelimgonzalez} Let $(\C,\W)$ and $(\C',\W')$ be two localizers and let $\begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] : G \end{tikzcd}$ be an adjunction 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 induces a natural transformation $\alpha : F' \circ \gamma \Rightarrow \gamma' \circ F$ defined as the following composition $\begin{tikzcd} & \C' \ar[rr,"\gamma'"]\ar[rd,"G"] & &\ho(\C') \ar[rr,"\mathrm{id}",""{name=B,below}]\ar[rd,"\RR G"'] & &\ho(\C') \\ \C\ar[ru,"F"] \ar[rr,"\mathrm{id}"',""{name=A,above}] && \C \ar[rr,"\gamma"'] &&\ho(\C)\ar[ru,"F'"'] &. \ar[from=A,to=1-2,"\eta",Rightarrow, shorten <= 0.5em, shorten >= 0.5em] \ar[from=2-3,to=1-4,Rightarrow,"\beta",shorten <= 1em, shorten >= 1em] \ar[from=2-5,to=B,Rightarrow,"\epsilon'"',shorten <= 0.5em, shorten >= 0.5em] \end{tikzcd}$ \end{paragr} \begin{proposition}[{\cite[Theorem 3.1]{gonzalez2012derivability}}]\label{prop:gonz} Let $(\C,\W)$ and $(\C',\W')$ be two localizers and $\begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] : G \end{tikzcd}$ be an adjunction. If $G$ is absolutely totally right derivable with $(\RR G,\beta)$ its left derived functor and if $\RR G$ has a left adjoint $F'$ $\begin{tikzcd} F' : \ho(\C) \ar[r,shift left] & \ho(\C') \ar[l,shift left] : \RR G, \end{tikzcd}$ then $F$ is absolutely totally left derivable and the pair $(F', \alpha)$, with $\alpha$ defined as in the previous paragraph, is its left derived functor. \end{proposition} % \todo{Gonzalez ne formule pas son théorème exactement de cette manière. Il % 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 $\underline{A}^{\op}$. 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 unique functor from $A$ to $e$ is denoted by $p_A : A \to e.$ \end{notation} \begin{definition} An \emph{op\nbd{}prederivator} is a (strict) $2$\nbd{}functor $\sD : \CCat^{\op} \to \CCAT.$ More explicitly, an op\nbd{}prederivator consists of the data of: \begin{itemize}[label=-] \item a big category $\sD(A)$ for every small category $A$, \item a functor $u^* : \sD(B) \to \sD(A)$ for every functor $u : A \to B$ between small categories, \item a natural transformation $\begin{tikzcd} \sD(B)\ar[r,bend left,"u^*",""{name=U,below}] \ar[r,bend right,"v^*"',""{name=D,above}] & \sD(A) \ar[from=U,to=D,Rightarrow,"\alpha^*"] \end{tikzcd}$ for every natural transformation $\begin{tikzcd} A \ar[r,bend left,"u",""{name=U,below}] \ar[r,bend right,"v"',""{name=D,above}] & B \ar[from=U,to=D,Rightarrow,"\alpha"] \end{tikzcd}$ with $A$ and $B$ small categories, \end{itemize} 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 $A$, $(1_A)^*=1_{\sD(A)}$, \item for every $u : A \to B$ and $v : B \to C$, $(vu)^*=u^* v^*$, \item for every $u : A \to B$, $(1_u)^*=1_{u^*}$, \item for every diagram in $\CCat$: $\begin{tikzcd}[column sep=large] A \ar[r,bend left=50, "u",""{name=U,below}] \ar[r,"v"description,""{name=V,above},""{name=W,below}] \ar[r,bend right=50,"w"',""{name=X,above}] & B, \ar[from=U,to=V,Rightarrow,"\alpha"] \ar[from=W,to=X,Rightarrow,"\beta"] \end{tikzcd}$ we have $(\alpha\beta)^*=\alpha^* \beta^*$, \item for every diagram in $\CCat$: $\begin{tikzcd}[column sep=large] A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B \ar[r,bend left,"u'",""{name=C,below}] \ar[r,bend right, "v'"',""{name=D,above}]&C, \ar[from=A,to=B,"\alpha",Rightarrow] \ar[from=C,to=D,"\beta",Rightarrow] \end{tikzcd}$ we have $(\beta \ast_0 \alpha)^*=\alpha^* \ast_0 \beta^*$. \end{itemize} \fi \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{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. \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 \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 $\CCat$, $u^* : \C(B) \to \C(A)$ is simply the functor induced from $u$ by pre-composition. \end{example} 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 $u^* : (\C(B),\W_B) \to (\C(A),\W_A)$ and every $\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B \ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ induces by pre-composition a $2$\nbd{}morphism of localizers $\begin{tikzcd} (\C(B),\W_B) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & (\C(A),\W_A). \ar[from=A,to=B,Rightarrow,"\alpha^*"] \end{tikzcd}$ (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 $u^*$, $u^* : \ho(\C(B)) \to \ho(\C(A))$ and every natural transformation $\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B \ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ induces a natural transformation, again denoted by $\alpha^*$, $\begin{tikzcd} \ho(\C(B)) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & \ho(\C(A)). \ar[from=A,to=B,Rightarrow,"\alpha^*"] \end{tikzcd}$ Altogether, this defines an op\nbd{}prederivator \begin{align*} \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 $\Ho(\C)(e)\simeq \ho(\C),$ 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 $u_! : \sD(A) \to \sD(B).$ \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 $p_A^* : \C \simeq \C(e) \to \C(A)$ is nothing but the diagonal functor that sends an object $X$ of $\C$ to the constant diagram with value $X$. Hence, the functor $p_{A!}$ is nothing but the usual colimit functor of $A$-shaped diagrams $p_{A!} = \colim_A : \C(A) \to \C(e) \simeq \C.$ \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 $\hocolim_A := p_{A!} : \ho(\C(A)) \to \ho(\C).$ For an object $X$ of $\ho(\C(A))$ (which is nothing but a diagram $X : A \to \C$ seen up to weak equivalence''), the object of $\ho(\C)$ $\hocolim_A(X)$ is the \emph{homotopy colimit of $X$}. For consistency, we also use the notation $\hocolim_{a \in A}X(a).$ When $\C$ is also cocomplete (which will always be the case in practice), it follows from Remark \ref{rem:homotopicalisder} and Proposition \ref{prop:gonz} that the functor $\colim_A : \C(A) \to \C$ is left derivable and $\hocolim_A$ is the left derived functor of $\colim_A$: $\LL \colim_A \simeq \hocolim_A.$ In particular, for every $A$-shaped diagram $X : A \to \C$, there is a canonical morphism of $\ho(\C)$ $\hocolim_A(X) \to \colim_A(X).$ This comparison map will be of great importance in the sequel. \end{paragr} \begin{paragr} Let $\begin{tikzcd} A \ar[r,"f"] \ar[d,"u"'] & B \ar[d,"v"]\\ C \ar[r,"g"'] & D \ar[from=1-2,to=2-1,Rightarrow,"\alpha"] \end{tikzcd}$ be a $2$\nbd{}square in $\CCat$. Every op\nbd{}prederivator $\sD$ induces a $2$\nbd{}square: $\begin{tikzcd} \sD(A) & \sD(B) \ar[l,"f^*"'] \\ \sD(C) \ar[u,"u^*"] & \sD(D). \ar[u,"v^*"'] \ar[l,"g^*"]\ar[from=1-2,to=2-1,Rightarrow,"\alpha^*"] \end{tikzcd}$ If $\sD$ has left Kan extensions, we obtain a canonical natural transformation $u_!f^* \Rightarrow g^*v_!$ referred to as the \emph{homological base change morphism induced by $\alpha$} and defined as the following composition: $\begin{tikzcd} \sD(C) & \ar[l,"u_!"'] \sD(A) & \sD(B) \ar[l,"f^*"'] \\ & \sD(C) \ar[ul,"\mathrm{id}",""{name=A,above}] \ar[u,"u^*"] & \sD(D) \ar[u,"v^*"'] \ar[l,"g^*"]& \sD(B). \ar[l,"v_!"]\ar[ul,"\mathrm{id}"',""{name=B,below}] \ar[from=1-3,to=2-2,Rightarrow,"\alpha^*"] \ar[from=1-2,to=A,Rightarrow,"\epsilon"'] \ar[from=B,to=2-3,Rightarrow,"\eta"] \end{tikzcd}$ In particular, let $u : A \to B$ be a morphism of $\CCat$ and $b$ an object of $B$ seen as a morphism $b :e \to B$. We have a square $\begin{tikzcd} A/b \ar[r,"k"] \ar[d,"p"']& A \ar[d,"u"] \\ e \ar[r,"b"'] & B \ar[from=1-2,to=2-1,Rightarrow,"\phi"] \end{tikzcd}$ where : \begin{itemize}[label=-] \item $A/b$ is the category whose objects are pairs $(a, f : u(a) \to b)$ with $a$ an object of $A$ and $f$ an arrow of $B$, and morphisms $(a,f) \to (a',f')$ are arrows $g : a \to a'$ of $A$ such that $f'\circ u(g) = f$, \item $k : A/b \to A$ is the functor $(a,p) \mapsto a$, \item $\phi$ is the natural transformation defined by $\phi_{(a,f)}:= f : u(a) \to b$. \end{itemize} Hence, we have a homological base change morphism: $p_!\, k^* \Rightarrow b^*u_!.$ In the case that $\sD$ is the homotopy op\nbd{}prederivator of a localizer $(\C,\W)$, for every object $X$ of $\sD(A)$ the above morphism reads $\hocolim_{A/b}(X\vert_{A/b}) \rightarrow u_!(X)_b$ where we use the notation $X\vert_{A/b}$ for $k^*(X)$ and $u_!(F)_b$ for $b^*(u_!(X))$. Note that this morphism is reminiscent of the formula that computes pointwise left Kan extensions in the classical'' sense (see for example \cite[chapter X, section 3]{mac2013categories}). %This formula is to be compare with formula \eqref{lknxtfrmla}. \end{paragr} \begin{definition}[Grothendieck] 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 $\sD(\amalg_{i \in I}A_i) \to \prod_{i \in I}\sD(A_i)$ is an equivalence of categories. In particular, $\sD(\emptyset)$ is equivalent to the terminal category. \item[Der 2)]\label{der2} For every small category $A$, the functor $\sD(A) \rightarrow \prod_{a \in \Ob(A)}\sD(e)$ induced by the functors $a^* : \sD(A)\to \sD(e)$ for all $a \in \Ob(A)$ (seen as morphisms $a : e \to A$), is conservative. \item[Der 3d)] $\sD$ admits left Kan extensions. \item[Der 4d)] For every $u : A \to B$ in $\CCat$ and $b$ object of $B$, the homological base change morphism $p_!\, k^* \Rightarrow b^*u_!$ induced by the square $\begin{tikzcd}[column sep=small, row sep=small] A/b \ar[r,"k"] \ar[d,"p"']& A \ar[d,"u"] \\ e \ar[r,"b"'] & B \ar[from=1-2,to=2-1,Rightarrow,"\phi"] \end{tikzcd}$ is an isomorphism. \end{description} \end{definition} \begin{paragr} Let us comment each of the axioms of the previous definition. Axiom \textbf{Der 1} ensures that $\sD(A)$ looks like'' a category of $A$-shaped diagrams. Axiom \textbf{Der 2} says that isomorphisms in $\sD(A)$ can be tested pointwise''. We have already seen that axiom \textbf{Der 3d} ensures the existence of left Kan extensions. Finally, axiom \textbf{Der 4d} intuitively says that Kan extensions are computed pointwise''. \end{paragr} \begin{example} Let $\C$ be a category. The op\nbd{}prederivator represented by $\C$ always satisfy axioms \textbf{Der 1} and \textbf{Der 2}. We have already seen that axioms \textbf{Der 3d} means exactly that $\C$ admits left Kan extensions in the classical sense, in which case axiom \textbf{Der 4d} is automatically satisfied. Hence, the op\nbd{}prederivator represented by $\C$ is a right op\nbd{}prederivator if and only if $\C$ is cocomplete. \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. \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. \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 \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. \end{paragr} \section{Morphisms of op-derivators, preservation of homotopy colimits} \sectionmark{Morphisms of op-derivators} We refer to \cite{leinster1998basic} for the precise definitions of pseudo-natural transformation (called strong transformation there) and 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: \begin{itemize}[label=-] \item a functor $F_A : \sD(A) \to \sD'(A)$ for every small category $A$, \item an isomorphism of functors $F_u: F_A u^* \overset{\sim}{\Rightarrow} u^* F_B$, $\begin{tikzcd} \sD(B) \ar[d,"u^*"] \ar[r,"F_B"] & \sD'(B) \ar[d,"u^*"]\\ \sD(A) \ar[r,"F_A"'] & \sD'(A), \ar[from=2-1,to=1-2,Rightarrow,"F_u","\sim"'] \end{tikzcd}$ for every $u : A \to B$ in $\CCat$. \end{itemize} These data being compatible with compositions and units. The morphism is \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. 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 $F : \C \to \C'$ 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 category $A$, the functor $F_A : \C(A) \to \C'(A)$ preserves weak equivalences and the universal property of the localization yields a functor $\overline{F}_A : \ho(\C(A)) \to \ho(\C'(A)).$ This defines a strict morphism of op\nbd{}prederivators: $\overline{F} : \Ho(\C) \to \Ho(\C').$ Similarly, every $2$\nbd{}morphism of localizers $\begin{tikzcd} (\C,\W) \ar[r,bend left,"F",""{name=A,below}] \ar[r,bend right, "G"',""{name=B,above}] &(\C',\W') \ar[from=A,to=B,"\alpha",Rightarrow] \end{tikzcd}$ induces a $2$\nbd{}morphism $\overline{\alpha} : \overline{F} \Rightarrow \overline{G}$. Altogether, we have defined a $2$\nbd{}functor \begin{align*} \Loc &\to \PPder\\ (\C,\W) &\mapsto \Ho(\C), \end{align*} 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 $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators. For every $u : A \to B$, there is a canonical natural transformation $u_!\, F_A \Rightarrow F_B\, u_!$ defined as $\begin{tikzcd} \sD(A) \ar[dr,"\mathrm{id}"',""{name=A,above}] \ar[r,"u_!"] &\sD(B) \ar[d,"u^*"] \ar[r,"F_B"] & \sD'(B)\ar[d,"u^*"] \ar[dr,"\mathrm{id}",""{name=B,below}]\\ & \sD(A) \ar[r,"F_A"'] & \sD'(A) \ar[r,"u_!"'] & \sD'(B). \ar[from=2-2,to=1-3,Rightarrow,"F_u"',"\sim"] \ar[from=A,to=1-2,Rightarrow,"\eta"] \ar[from=2-3,to=B,Rightarrow,"\epsilon"] \end{tikzcd}$ 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 $\hocolim_{A}(F_A(X))\to F_e(\hocolim_A(X)).$ \end{paragr} \begin{definition}\label{def:cocontinuous} Let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators and suppose that $\sD$ and $\sD'$ both admit left Kan extensions. We say that $F$ is \emph{cocontinuous}\footnote{Some authors also say \emph{left exact}.} if for every $u: A \to B$, the canonical morphism $u_! \, F_A \Rightarrow F_B \, u_!$ 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 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. \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: \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 $\mathrm{id}_{\sD}$; 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 $F$) 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. \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, resp.\ op-derivator) if and only if $\sD'$ is one. \end{lemma} \begin{lemma}\label{lemma:eqisadj} Let $F : \sD \to \sD'$ be an equivalence and $G : \sD' \to \sD$ be a 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. \end{lemma} We end this section with a generalization of the notion of localization in the context of op\nbd{}prederivators. \begin{paragr} Let $(\C,\W)$ be a localizer. For every small category $A$, let $\gamma_A : \C(A) \to \ho(\C(A))$ be the localization functor. The correspondence $A \mapsto \gamma_A$ is natural in $A$ and defines a strict morphism of op\nbd{}prederivators $\gamma : \C \to \Ho(\C).$ \end{paragr} \begin{definition}\label{def:strnglyder} Let $(\C,\W)$ and $(\C',\W')$ be two localizers and $F : \C \to \C'$ a functor. We say that $F$ is \emph{strongly left derivable} if there exists a morphism of op\nbd{}prederivators $\LL F : \Ho(\C) \to \Ho(\C')$ and a $2$\nbd{}morphism of op\nbd{}prederivators $\begin{tikzcd} \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\ \Ho(\C) \ar[r,"\LL F"'] & \Ho(\C'). \arrow[from=2-1, to=1-2,"\alpha",Rightarrow] \end{tikzcd}$ 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. \end{definition} \begin{example} Let $(\C,\W)$ and $(\C,\W')$ be localizers and $F : \C \to \C'$ a functor. If $F$ preserves weak equivalences (i.e.\ it is a morphism of localizers), then it is strongly left and right derivable and $\overline{F} \simeq \LL F \simeq \RR G.$ \end{example} Gonzalez' criterion (Proposition \ref{prop:gonz}) admits the following generalization. \begin{proposition}\label{prop:gonzalezcritder} Let $(\C,\W)$ and $(\C',\W')$ be two localizers and $\begin{tikzcd} F : \C \ar[r,shift left] & \C' : G \ar[l,shift left] \end{tikzcd}$ be an adjunction. If $G$ is strongly right derivable and if $\mathbb{R}G$ has a left adjoint $F'$ $\begin{tikzcd} F' : \Ho(\C) \ar[r,shift left] & \Ho(\C') : \RR G, \ar[l,shift left] \end{tikzcd}$ then $F$ is strongly left derivable and $\LL F \simeq F'.$ \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. Proposition \ref{prop:gonz} gives us that for every small category $A$, the functor $F_A$ is absolutely totally left derivable with $(F'_A,\alpha_A)$ 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 $F$. \end{proof} \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 $\begin{tikzcd} (0,0) \ar[r] \ar[d] & (0,1)\ar[d] \\ (1,0) \ar[r] & (1,1) \end{tikzcd}$ and we use the notation $\ulcorner$ for the full subcategory of $\square$ spanned by $(0,0)$, $(0,1)$ and $(1,0)$, which can be pictured as $\begin{tikzcd} (0,0) \ar[d] \ar[r] & (0,1) \\ (1,0) &. \end{tikzcd}$ Finally, we write $i_{\ulcorner} : \ulcorner \to \square$ for the canonical inclusion functor. \end{paragr} \begin{definition}\label{def:cocartesiansquare} Let $\sD$ be an op\nbd{}prederivator. An object $X$ of $\sD(\square)$ is \emph{cocartesian} if for every object $Y$ of $\sD(\square)$, the canonical map $\Hom_{\sD(\square)}(X,Y) \to \Hom_{\sD(\ulcorner)}(i_{\ulcorner}^*(X),i_{\ulcorner}^*(Y))$ induced by the functor $i_{\ulcorner}^* : \sD(\square) \to \sD(\ulcorner)$, is a bijection. \end{definition} \begin{example} Let $\C$ be a category. An object of $\C(\square)$ is nothing but a commutative square in $\C$ and it is cocartesian in the sense of the previous definition if and only if it is cocartesian in the usual sense. \end{example} For the following definition to make sense, recall that for a localizer $(\C,\W)$ and a small category $A$, the objects of $\Ho(\C)(A)=\ho(\C(A))$ are identified with the objects of $\C(A)$ via the localization functor. In particular, an object of $\Ho(\C)(\square)$ is a commutative square of $\C$ (up to weak equivalence). \begin{definition}\label{def:hmtpycocartesiansquare} Let $(\C,\W)$ be a localizer. A commutative square of $\C$ is said to be \emph{homotopy cocartesian} if it is cocartesian in $\Ho(\C)$ in the sense of Definition \ref{def:cocartesiansquare}. \end{definition} \begin{paragr} Let $\sD$ be an op\nbd{}prederivator. The object $(1,1)$ of $\square$ can be considered as a morphism of $\Cat$ $(1,1) : e \to \square$ and thus induces a functor $(1,1)^* : \sD(\square)\to \sD(e)$. For an object $X$ of $\sD(\square)$, we use the notation $X_{(1,1)} := (1,1)^*(X).$ Now, since $(1,1)$ is the terminal object of $\square$, we have a canonical $2$\nbd{}triangle $\begin{tikzcd} \ulcorner \ar[rr,"i_{\ulcorner}",""{name=A,below}] \ar[rd,"p"'] && \square\\ &e\ar[ru,"{(1,1)}"']&, \ar[from=A,to=2-2,Rightarrow,"\alpha"] \end{tikzcd}$ where we wrote $p$ instead of $p_{\ulcorner}$ for short. Hence, we have a $2$\nbd{}triangle $\begin{tikzcd} \sD(\ulcorner) && \sD(\square) \ar[ll,"i_{\ulcorner}^*"',""{name=A,below}] \ar[dl,"{(1,1)}^*"] \\ & \sD(e) \ar[ul,"p^*"]&. \ar[from=A,to=2-2,Rightarrow,"\alpha^*"] \end{tikzcd}$ Suppose now that $\sD$ has left Kan extensions. For $X$ an object of $\sD(\square)$, we have a canonical morphism $p_!(i_{\ulcorner}^*(X)) \to X_{(1,1)}$ defined as the composition $p_{!}(i_{\ulcorner}^*(X)) \to p_{!} p^* (X_{(1,1)}) \to X_{(1,1)},$ where the arrow on the left is induced by $\alpha^*$ and the arrow on the right is induced by the co-unit of the adjunction $p_{!} \dashv p^*$. When $\sD$ is the homotopy op\nbd{}prederivator of a localizer, and $X$ is a commutative square of $\C$ $X= \begin{tikzcd} A \ar[r,"u"] \ar[d,"f"]& B \ar[d,"g"] \\ C \ar[r,"v"]&D, \end{tikzcd}$ this previous morphism reads $\hocolim\left(\begin{tikzcd}[column sep=tiny, row sep=tiny] A \ar[d,"f"] \ar[r,"u"] & B \\ C & \end{tikzcd}\right) \to D.$ \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. \end{proposition} \begin{proof} Let $Y$ be another object of $\sD(\square)$. Using the adjunction $i_{\ulcorner!} \dashv i_{\ulcorner}^*$, the canonical map $\Hom_{\sD(\square)}(X,Y) \to \Hom_{\sD(\ulcorner)}(i_{\ulcorner}^*(X),i_{\ulcorner}^*(Y))$ can be identified with $\Hom_{\sD(\square)}(X,Y) \to \Hom_{\sD(\ulcorner)}(i_{\ulcorner !}(i_{\ulcorner}^*(X)),Y).$ Hence, $X$ is cocartesian if and only if the co-unit map $X \to i_{\ulcorner !}(i_{\ulcorner}^*(X))$ 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 commutative square of $\C$ is homotopy cocartesian if and only if the bottom right apex of the square is the homotopy colimit of the upper left corner of the square. This hopefully justifies the terminology of cocartesian square''. The previous proposition admits the following immediate corollary. \begin{corollary} Let $\sD$ and $\sD'$ be right op-derivators and $F : \sD \to \sD'$ a morphism of op\nbd{}prederivators. If $F$ is cocontinuous, then it preserves cocartesian squares. This means that if an object $X$ of $\sD(\square)$ is cocartesian, then $F_{\square}(X)$ is a cocartesian square of $\sD'$. \end{corollary} We end this section with two useful lemmas which show that homotopy cocartesian squares behave much like classical'' cocartesian squares. \begin{lemma}\label{lemma:hmtpycocartsquarewe} Let $(\C,\W)$ be a homotopy cocomplete localizer and let $\begin{tikzcd} A \ar[r,"u"] \ar[d,"f"]& B \ar[d,"g"] \\ C \ar[r,"v"]&D \end{tikzcd}$ be a commutative square in $\C$. If $f$ and $g$ are weak equivalences then the previous square is homotopy cocartesian. \end{lemma} \begin{proof} Using \textbf{Der 2}, one can show that the previous square is isomorphic in $\Ho(\C)(\square)$ to the square $\begin{tikzcd} A \ar[r,"u"] \ar[d,"1_A"]& B \ar[d,"1_B"] \\ A \ar[r,"u"]&B. \end{tikzcd}$ The result follows then from \cite[Proposition 3.12(2)]{groth2013derivators}. \end{proof} \begin{lemma}[Pasting lemma]\label{lemma:pastinghmtpycocartesian} Let $(\C,\W)$ be a homotopy cocomplete localizer and let $\begin{tikzcd} A \ar[r,"u"] \ar[d,"f"]& B \ar[d,"g"] \ar[r,"v"] & C \ar[d,"h"]\\ D \ar[r,"w"]&E \ar[r,"x"] & F \end{tikzcd}$ be a commutative diagram in $\C$. If the square on the left is cocartesian then the outer square is cocartesian if and only if the right square is cocartesian. \end{lemma} \begin{proof} This is a particular case of \cite[Proposition 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,hirschhorn2009model,dwyer1995homotopy}) for basic definitions and results. For a model category $\M = (\M,\W,\Cof,\Fib)$, the homotopy op\nbd{}prederivator of $\M$, denoted by $\Ho(\M)$, is the homotopy op\nbd{}prederivator of the localizer $(\M,\W)$. The following theorems are due to Cisinski \cite{cisinski2003images} and can be summed up by the slogan: \begin{center} Model categories are homotopy cocomplete and left Quillen functors are homotopy cocontinuous. \end{center} \begin{theorem}[Cisinski]\label{thm:cisinskiI} Let $(\M,\W,\Cof,\Fib)$ be a model category. The localizer $(\M,\W)$ is homotopy cocomplete. \end{theorem} \begin{theorem}[Cisinski]\label{thm:cisinskiII} Let $\M$ and $\M'$ be model categories. Let $F \colon \M \to \M'$ be 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 $\LL F \colon \Ho(\M) \to \Ho(\M')$ is homotopy cocontinuous. \end{theorem} \begin{remark} The obvious duals of the two above theorems are also true. The reason we put emphasis on cocompleteness rather that completeness is because we will make no use whatsoever of homotopy limits in this dissertation. \end{remark} \begin{remark} Note that for a model category $(\M,\W,\Cof,\Fib)$, its homotopy op\nbd{}prederivator only depends on its underlying localizer. Hence, the existence of the classes $\Cof$ and $\Fib$ with the usual properties defining a model structure ought to be thought of as a \emph{property} of the localizer $(\M,\W)$, which is sufficient 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 $(\M,\W,\Cof,\Fib)$ has 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 \begin{aligned} X \otimes a : A &\to \C \\ b &\mapsto \coprod_{\Hom_A(a,b)}X. \end{aligned} For every object $a$ of $A$, this gives rise to a functor \begin{aligned} \shortminus \otimes a : \C &\to \C(A)\\ X &\mapsto X \otimes a. \end{aligned} \end{paragr} \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: \begin{itemize}[label=-] \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 cofibrations (resp.\ trivial cofibrations) is given by $\{ f \otimes a : X \otimes a \to Y \otimes a \quad \vert \quad a \in \Ob(A), f \in I\}$ (resp. $\{ f \otimes a : X \otimes a \to Y \otimes a \quad \vert \quad a \in \Ob(A), f \in J\}).$ \end{proposition} \begin{proof} See for example \cite[Proposition 3.4]{cisinski2003images}. \end{proof} \begin{paragr}\label{paragr:projmod} The model structure of the previous proposition is referred to as the \emph{projective model structure on $\M(A)$}. \end{paragr} \begin{proposition} Let $(\M,\W,\Cof,\Fib)$ be a cofibrantly generated model category. For every $u : A \to B$, the adjunction $\begin{tikzcd} u_! : \M(A) \ar[r,shift left]& \ar[l,shift left] \M(B) : u^* \end{tikzcd}$ is a Quillen adjunction with respect to the projective model structures on $\M(A)$ and on $\M(B)$. \end{proposition} \begin{proof} By definition of the projective model structure, $u^*$ preserve weak equivalences and fibrations. \end{proof} \begin{paragr}\label{paragr:hocolimms} In particular, in the case that $B$ is the terminal category $e$, we have that $\begin{tikzcd} \colim_A : \M(A) \ar[r,shift left] & \ar[l,shift left] \M(e) \simeq \M : p_A^* \end{tikzcd}$ is a Quillen adjunction. Since $\hocolim_A$ is the left derived functor of $\colim_A$, we obtain the following immediate corollary of the previous proposition. \end{paragr} \begin{corollary}\label{cor:cofprojms} Let $(\M,\W,\Cof,\Fib)$ be a cofibrantly generated model category, $A$ a small category and $X : A \to \M$ a diagram. If $X$ is cofibrant for the projective model structure on $\C(A)$, then the canonical morphism of $\ho(\C)$ $\hocolim_A(X) \to \colim_A(X)$ is an isomorphism. \end{corollary} \begin{proof} This is simply a particular case of the general fact that for a left Quillen functor $F : \M \to \M'$ and a cofibrant object $X$ of $\M$, the canonical map of $\ho(\M')$ $\LL F(X) \to F(X)$ is an isomorphism. \end{proof} Below is a particular case for which the previous corollary applies. \begin{proposition}\label{prop:sequentialhmtpycolimit} Let $(\M,\W,\Cof,\Fib)$ be a cofibrantly generated model category and let $X$ be a sequential diagram in $\M$ $X_0 \to X_1 \to X_2 \to \cdots$ (i.e.\ a diagram $X : (\mathbb{N},\leq) \to \M$). If $X_0$ is cofibrant and each $X_i \to X_{i+1}$ is a cofibration, then $X$ is cofibrant for the projective model structure on $\M((\mathbb{N},\leq))$. \end{proposition} \begin{proof} It is an easy exercise that uses only the fact that cofibrations of the projective model structure are, by definition, the morphisms with left lifting property to pointwise fibrations. For details see \cite[Example 2.3.16]{schreiber2013differential}. \end{proof} Another setting for which a category of diagrams $\M(A)$ can be equipped with a model structure whose weak equivalences are the pointwise equivalences and for which the $A$-colimit functor is left Quillen is when the category $A$ is a \emph{Reedy category}. Rather that recalling this theory, we simply put here the only practical result that we shall need in the sequel. \begin{lemma}\label{lemma:hmtpycocartesianreedy} Let $(\M,\W,\Cof,\Fib)$ be a model category and let $\begin{tikzcd} A \ar[r,"u"] \ar[d,"f"]& B \ar[d,"g"] \\ C \ar[r,"v"]&D \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd}$ be a \emph{cocartesian} square in $\M$. If either $u$ or $f$ is a cofibration and if $A$, $B$ and $C$ are cofibrant objects, then this square is \emph{homotopy cocartesian}. \end{lemma} \begin{proof} See for example \cite[Proposition A.2.4.4(i)]{lurie2009higher}. \end{proof} %%% Local Variables: %%% mode: latex %%% TeX-master: "main" %%% End: