homtheo.tex 48 KB
 Leonard Guetta committed Jun 25, 2020 1 \chapter{Homotopical algebra}  Leonard Guetta committed Oct 23, 2020 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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and in the first section of \cite{cisinski2003images}; finally, a rather complete (yet unfinished and unpublished) textbook on the subject is \cite{groth2013book}.  Leonard Guetta committed Aug 20, 2020 16   Leonard Guetta committed Oct 23, 2020 17 18 \iffalse Let us quickly motive this choice for the reader unfamiliar with this theory.  Leonard Guetta committed Aug 16, 2020 19   Leonard Guetta committed Oct 23, 2020 20 21 22 23 24 25 26 27 28 29 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  Leonard Guetta committed Jun 25, 2020 30 31 32 \section{Localization, derivation} \begin{paragr}\label{paragr:loc}  Leonard Guetta committed Oct 23, 2020 33 34 35 36  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  Leonard Guetta committed Jun 25, 2020 37  $ Leonard Guetta committed Oct 23, 2020 38  \gamma : \C \to \ho(\C)  Leonard Guetta committed Jun 25, 2020 39 $  Leonard Guetta committed Oct 23, 2020 40  the localization functor \cite[1.1]{gabriel1967calculus}. Recall the universal  Leonard Guetta committed Oct 25, 2020 41  property of the localization: for every category $\D$, the functor induced by  Leonard Guetta committed Oct 23, 2020 42  pre-composition  Leonard Guetta committed Jun 25, 2020 43  $ Leonard Guetta committed Oct 23, 2020 44  \gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D)  Leonard Guetta committed Jun 25, 2020 45 $  Leonard Guetta committed Oct 23, 2020 46  is fully faithful and its essential image consists of functors $F~:~\C~\to~\D$  Leonard Guetta committed Dec 27, 2020 47  that send the morphisms of $\W$ to isomorphisms of $\D$.  Leonard Guetta committed Jun 25, 2020 48   Leonard Guetta committed Oct 23, 2020 49 50  We shall always consider that $\C$ and $\ho(\C)$ have the same class of objects and implicitly use the equality  Leonard Guetta committed Jun 25, 2020 51  $ Leonard Guetta committed Oct 23, 2020 52  \gamma(X)=X  Leonard Guetta committed Jun 25, 2020 53 $  Leonard Guetta committed Oct 24, 2020 54  for every object $X$ of $\C$.  Leonard Guetta committed Aug 16, 2020 55   Leonard Guetta committed Oct 23, 2020 56  The class of arrows $\W$ is said to be \emph{saturated} when we have the  Leonard Guetta committed Oct 24, 2020 57  property:  Leonard Guetta committed Aug 16, 2020 58  $ Leonard Guetta committed Oct 23, 2020 59  f \in \W \text{ if and only if } \gamma(f) \text{ is an isomorphism. }  Leonard Guetta committed Aug 16, 2020 60 $  Leonard Guetta committed Jun 25, 2020 61 \end{paragr}  Leonard Guetta committed Aug 21, 2020 62 63 For later reference, we put here the following definition. \begin{definition}\label{def:couniversalwe}  Leonard Guetta committed Oct 23, 2020 64 65 66  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  Leonard Guetta committed Aug 21, 2020 67  $ Leonard Guetta committed Oct 23, 2020 68 69  \begin{tikzcd} X \ar[r] \ar[d,"f"] & X' \ar[d,"f'"] \\  Leonard Guetta committed Oct 24, 2020 70  Y \ar[r] & Y', \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Aug 21, 2020 71 72  \end{tikzcd}$  Leonard Guetta committed Oct 23, 2020 73 74  the morphism $f'$ is also a weak equivalence. \end{definition}  Leonard Guetta committed Jun 25, 2020 75 \begin{paragr}  Leonard Guetta committed Oct 23, 2020 76  A \emph{morphism of localizers} $F : (\C,\W) \to (\C',\W')$ is a functor  Leonard Guetta committed Oct 30, 2020 77 78 79  $F:\C\to\C'$ 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  Leonard Guetta committed Jun 25, 2020 80  $ Leonard Guetta committed Oct 23, 2020 81  \overline{F} : \ho(\C) \to \ho(\C')  Leonard Guetta committed Jun 25, 2020 82 83 $ such that the square  Leonard Guetta committed Oct 23, 2020 84 85 86 87 88  $\begin{tikzcd} \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\ \ho(\C) \ar[r,"\overline{F}"] & \ho(\C'). \end{tikzcd}  Leonard Guetta committed Jun 25, 2020 89 $  Leonard Guetta committed Oct 23, 2020 90  is commutative. Let $G : (\C,\W) \to (\C',\W')$ be another morphism of  Leonard Guetta committed Oct 30, 2020 91 92  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  Leonard Guetta committed Oct 27, 2020 93  the localization implies that there exists a unique natural transformation  Leonard Guetta committed Jun 25, 2020 94  $ Leonard Guetta committed Oct 23, 2020 95 96 97  \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}  Leonard Guetta committed Jun 25, 2020 98 $  Leonard Guetta committed Oct 24, 2020 99  such that the $2$\nbd{}diagram  Leonard Guetta committed Jun 25, 2020 100  $ Leonard Guetta committed Oct 23, 2020 101 102 103 104 105 106  \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}  Leonard Guetta committed Jun 25, 2020 107 108 109 $ is commutative in an obvious sense. \end{paragr}  Leonard Guetta committed Sep 04, 2020 110 \begin{remark}\label{remark:localizedfunctorobjects}  Leonard Guetta committed Oct 30, 2020 111 112  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  Leonard Guetta committed Jan 20, 2021 113 114  is the identity on objects, it follows that for a morphism of localizers $F \colon (\C,\W) \to (\C',\W')$, we tautologically have  Leonard Guetta committed Sep 04, 2020 115  $ Leonard Guetta committed Oct 23, 2020 116  \overline{F}(X)=F(X)  Leonard Guetta committed Sep 04, 2020 117 118 119 $ for every object $X$ of $\C$. \end{remark}  Leonard Guetta committed Jun 25, 2020 120 \begin{paragr}\label{paragr:defleftderived}  Leonard Guetta committed Oct 23, 2020 121 122  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  Leonard Guetta committed Jun 25, 2020 123  $ Leonard Guetta committed Oct 23, 2020 124  \LL F : \ho(\C) \to \ho(\C')  Leonard Guetta committed Jun 25, 2020 125 $  Leonard Guetta committed Oct 23, 2020 126  and a natural transformation  Leonard Guetta committed Jun 25, 2020 127  $ Leonard Guetta committed Oct 23, 2020 128  \alpha : \LL F \circ \gamma \Rightarrow \gamma'\circ F  Leonard Guetta committed Jun 25, 2020 129 $  Leonard Guetta committed Oct 23, 2020 130 131  that makes $\LL F$ the \emph{right} Kan extension of $\gamma' \circ F$ along $\gamma$:  Leonard Guetta committed Jun 25, 2020 132  $ Leonard Guetta committed Oct 23, 2020 133 134 135 136 137  \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}  Leonard Guetta committed Jun 25, 2020 138 $  Leonard Guetta committed Oct 23, 2020 139 140 141 142 143 144  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 $F$.  Leonard Guetta committed Jun 25, 2020 145   Leonard Guetta committed Oct 30, 2020 146 147  The notion of \emph{total right derivable functor} is defined dually and denoted by $\RR F$ when it exists.  Leonard Guetta committed Jun 25, 2020 148 149 \end{paragr} \begin{example}\label{rem:homotopicalisder}  Leonard Guetta committed Oct 23, 2020 150 151 152 153 154  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 $\LL F \simeq \RR F \simeq \overline{F}$.  Leonard Guetta committed Jun 25, 2020 155 \end{example}  Leonard Guetta committed Oct 23, 2020 156 157 To end this section, we recall a derivability criterion due to Gonzalez, which we shall use in the sequel.  Leonard Guetta committed Aug 21, 2020 158 \begin{paragr}\label{paragr:prelimgonzalez}  Leonard Guetta committed Oct 23, 2020 159 160 161 162 163  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  Leonard Guetta committed Oct 30, 2020 164  adjunction being denoted by $\epsilon'$. All this data induces a natural  Leonard Guetta committed Oct 23, 2020 165 166  transformation $\alpha : F' \circ \gamma \Rightarrow \gamma' \circ F$ defined as the following composition  Leonard Guetta committed Aug 21, 2020 167  $ Leonard Guetta committed Oct 23, 2020 168 169 170 171 172 173 174 175 176  \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}  Leonard Guetta committed Aug 21, 2020 177 178 $ \end{paragr}  Leonard Guetta committed Oct 23, 2020 179 180 \begin{proposition}[{\cite[Theorem 3.1]{gonzalez2012derivability}}]\label{prop:gonz}  Leonard Guetta committed Jun 25, 2020 181  Let $(\C,\W)$ and $(\C',\W')$ be two localizers and  Leonard Guetta committed Oct 23, 2020 182  $\begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] :  Leonard Guetta committed Oct 24, 2020 183  G \end{tikzcd}$ be an adjunction. If $G$ is absolutely totally right  Leonard Guetta committed Oct 23, 2020 184 185 186 187 188 189 190 191 192  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.}  Leonard Guetta committed Jun 25, 2020 193   Leonard Guetta committed Oct 26, 2020 194 \section{(op-)Derivators and homotopy colimits}  Leonard Guetta committed Oct 30, 2020 195 196 197 198 199 \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 $\underline{A}^{\op}$.  Leonard Guetta committed Jun 25, 2020 200   Leonard Guetta committed Oct 23, 2020 201 202  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 $A$,  Leonard Guetta committed Oct 30, 2020 203  the unique functor from $A$ to $e$ is denoted by  Leonard Guetta committed Jun 25, 2020 204  $ Leonard Guetta committed Oct 23, 2020 205  p_A : A \to e.  Leonard Guetta committed Jun 25, 2020 206 207 208 $ \end{notation} \begin{definition}  Leonard Guetta committed Oct 25, 2020 209  An \emph{op\nbd{}prederivator} is a (strict) $2$\nbd{}functor  Leonard Guetta committed Oct 29, 2020 210  $\sD : \CCat^{\op} \to \CCAT.$  Leonard Guetta committed Oct 25, 2020 211  More explicitly, an op\nbd{}prederivator consists of the data of:  Leonard Guetta committed Jun 25, 2020 212 213  \begin{itemize}[label=-] \item a big category $\sD(A)$ for every small category $A$,  Leonard Guetta committed Oct 23, 2020 214 215  \item a functor $u^* : \sD(B) \to \sD(A)$ for every functor $u : A \to B$ between small categories,  Leonard Guetta committed Jun 25, 2020 216 217  \item a natural transformation $ Leonard Guetta committed Oct 23, 2020 218 219 220 221  \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^*"]  Leonard Guetta committed Jun 25, 2020 222 223 224 225  \end{tikzcd}$ for every natural transformation $ Leonard Guetta committed Oct 23, 2020 226 227 228  \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"]  Leonard Guetta committed Jun 25, 2020 229  \end{tikzcd}  Leonard Guetta committed Oct 23, 2020 230 231 $ with $A$ and $B$ small categories,  Leonard Guetta committed Jun 25, 2020 232  \end{itemize}  Leonard Guetta committed Oct 30, 2020 233  compatible with compositions and units (in a strict sense).\iffalse such that  Leonard Guetta committed Oct 23, 2020 234  the following axioms are satisfied:  Leonard Guetta committed Jun 25, 2020 235 236 237 238 239 240  \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 $\CCat$: $ Leonard Guetta committed Oct 23, 2020 241 242 243 244 245  \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"]  Leonard Guetta committed Jun 25, 2020 246 247 248 249 250  \end{tikzcd}$ we have $(\alpha\beta)^*=\alpha^* \beta^*$, \item for every diagram in $\CCat$: $ Leonard Guetta committed Oct 23, 2020 251 252 253 254 255 256  \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}  Leonard Guetta committed Jun 25, 2020 257 258 259 260 261 $ we have $(\beta \ast_0 \alpha)^*=\alpha^* \ast_0 \beta^*$. \end{itemize} \fi \end{definition}  Leonard Guetta committed Aug 16, 2020 262 \begin{remark}  Leonard Guetta committed Oct 23, 2020 263  Note that some authors call \emph{prederivator} what we have called  Leonard Guetta committed Oct 30, 2020 264 265  \emph{op\nbd{}prederivator}. The terminology we chose in the above definition is compatible with the original one of Grothendieck, who called  Leonard Guetta committed Oct 24, 2020 266  \emph{prederivator} a $2$\nbd{}functor from $\CCat$ to $\CCAT$ that is  Leonard Guetta committed Oct 30, 2020 267 268  contravariant at the level of $1$-cells \emph{and} at the level of $2$\nbd{}cells.  Leonard Guetta committed Aug 16, 2020 269 \end{remark}  Leonard Guetta committed Jun 25, 2020 270 \begin{example}\label{ex:repder}  Leonard Guetta committed Oct 23, 2020 271 272 273  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  Leonard Guetta committed Oct 30, 2020 274 275  $2$\nbd{}functorial in an obvious sense and thus defines an op\nbd{}prederivator  Leonard Guetta committed Jun 25, 2020 276  \begin{align*}  Leonard Guetta committed Oct 29, 2020 277  \C : \CCat^{\op} &\to \CCAT \\  Leonard Guetta committed Jun 25, 2020 278  A &\mapsto \C(A)  Leonard Guetta committed Oct 23, 2020 279  \end{align*}  Leonard Guetta committed Oct 30, 2020 280 281  which we call the op\nbd{}prederivator \emph{represented by $\C$}. For $u : A \to B$ in $\CCat$,  Leonard Guetta committed Oct 23, 2020 282  $ Leonard Guetta committed Dec 27, 2020 283  u^* : \C(B) \to \C(A)  Leonard Guetta committed Oct 23, 2020 284 285 $ is simply the functor induced from $u$ by pre-composition.  Leonard Guetta committed Jun 25, 2020 286 \end{example}  Leonard Guetta committed Aug 16, 2020 287   Leonard Guetta committed Oct 25, 2020 288 We now turn to the most important way of obtaining op\nbd{}prederivators.  Leonard Guetta committed Jun 25, 2020 289 \begin{paragr}\label{paragr:homder}  Leonard Guetta committed Oct 30, 2020 290 291 292 293 294 295 296  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  Leonard Guetta committed Jun 25, 2020 297  $ Leonard Guetta committed Oct 23, 2020 298  u^* : (\C(B),\W_B) \to (\C(A),\W_A)  Leonard Guetta committed Jun 25, 2020 299 $  Leonard Guetta committed Oct 23, 2020 300 301 302  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  Leonard Guetta committed Oct 24, 2020 303  a $2$\nbd{}morphism of localizers  Leonard Guetta committed Jun 25, 2020 304  $ Leonard Guetta committed Oct 23, 2020 305 306 307 308 309  \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}  Leonard Guetta committed Jun 25, 2020 310 $  Leonard Guetta committed Oct 30, 2020 311 312 313 314  (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 $u^*$,  Leonard Guetta committed Jun 25, 2020 315  $ Leonard Guetta committed Oct 23, 2020 316  u^* : \ho(\C(B)) \to \ho(\C(A))  Leonard Guetta committed Jun 25, 2020 317 $  Leonard Guetta committed Oct 30, 2020 318 319  and every natural transformation $\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B  Leonard Guetta committed Dec 27, 2020 320  \ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ induces a natural  Leonard Guetta committed Oct 25, 2020 321  transformation, again denoted by $\alpha^*$,  Leonard Guetta committed Jun 25, 2020 322 323  $\begin{tikzcd}  Leonard Guetta committed Oct 23, 2020 324 325 326 327  \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}  Leonard Guetta committed Jun 25, 2020 328 $  Leonard Guetta committed Oct 25, 2020 329  Altogether, this defines an op\nbd{}prederivator  Leonard Guetta committed Jun 25, 2020 330  \begin{align*}  Leonard Guetta committed Oct 29, 2020 331  \Ho^{\W}(\C) : \CCat^{\op} &\to \CCAT\\  Leonard Guetta committed Jun 25, 2020 332 333  A &\mapsto \ho(\C(A)), \end{align*}  Leonard Guetta committed Oct 30, 2020 334 335 336 337 338  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  Leonard Guetta committed Jun 25, 2020 339  $ Leonard Guetta committed Oct 25, 2020 340  \Ho(\C)(e)\simeq \ho(\C),  Leonard Guetta committed Jun 25, 2020 341 $  Leonard Guetta committed Oct 25, 2020 342  which we shall use without further reference.  Leonard Guetta committed Jun 25, 2020 343 \end{paragr}  Leonard Guetta committed Aug 17, 2020 344 \begin{definition}  Leonard Guetta committed Oct 30, 2020 345 346  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  Leonard Guetta committed Jun 25, 2020 347  $ Leonard Guetta committed Oct 23, 2020 348  u_! : \sD(A) \to \sD(B).  Leonard Guetta committed Jun 25, 2020 349 $  Leonard Guetta committed Aug 17, 2020 350 351 \end{definition} \begin{example}  Leonard Guetta committed Oct 30, 2020 352 353 354 355 356 357  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  Leonard Guetta committed Jun 25, 2020 358  $ Leonard Guetta committed Aug 17, 2020 359  p_A^* : \C \simeq \C(e) \to \C(A)  Leonard Guetta committed Oct 23, 2020 360 361 362 $ 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  Leonard Guetta committed Dec 27, 2020 363  the usual colimit functor of $A$-shaped diagrams  Leonard Guetta committed Oct 23, 2020 364  $ Leonard Guetta committed Aug 17, 2020 365  p_{A!} = \colim_A : \C(A) \to \C(e) \simeq \C.  Leonard Guetta committed Oct 23, 2020 366 $  Leonard Guetta committed Jun 25, 2020 367 368 \end{example} \begin{paragr}  Leonard Guetta committed Oct 30, 2020 369 370 371 372  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  Leonard Guetta committed Jun 25, 2020 373  $ Leonard Guetta committed Oct 23, 2020 374  \hocolim_A := p_{A!} : \ho(\C(A)) \to \ho(\C).  Leonard Guetta committed Jun 25, 2020 375 $  Leonard Guetta committed Oct 24, 2020 376  For an object $X$ of $\ho(\C(A))$ (which is nothing but a diagram $X : A \to  Leonard Guetta committed Oct 25, 2020 377  \C$ seen up to weak equivalence''), the object of $\ho(\C)$  Leonard Guetta committed Jun 25, 2020 378  $ Leonard Guetta committed Oct 23, 2020 379  \hocolim_A(X)  Leonard Guetta committed Jun 25, 2020 380 $  Leonard Guetta committed Oct 23, 2020 381 382  is the \emph{homotopy colimit of $X$}. For consistency, we also use the notation  Leonard Guetta committed Aug 17, 2020 383 384  $\hocolim_{a \in A}X(a).  Leonard Guetta committed Oct 23, 2020 385 $  Leonard Guetta committed Oct 25, 2020 386  When $\C$ is also cocomplete (which will always be the case in practice), it  Leonard Guetta committed Oct 23, 2020 387 388 389 390 391  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 $\colim_A$: $ Leonard Guetta committed Jun 25, 2020 392  \LL \colim_A \simeq \hocolim_A.  Leonard Guetta committed Oct 23, 2020 393 394 395 396 $ In particular, for every $A$-shaped diagram $X : A \to \C$, there is a canonical morphism of $\ho(\C)$ $ Leonard Guetta committed Aug 17, 2020 397  \hocolim_A(X) \to \colim_A(X).  Leonard Guetta committed Oct 23, 2020 398 399 $ This comparison map will be of great importance in the sequel.  Leonard Guetta committed Jun 25, 2020 400 \end{paragr}  Leonard Guetta committed Oct 23, 2020 401 402 \begin{paragr} Let  Leonard Guetta committed Jun 25, 2020 403  $ Leonard Guetta committed Oct 23, 2020 404 405 406 407  \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}  Leonard Guetta committed Jun 25, 2020 408 $  Leonard Guetta committed Oct 30, 2020 409 410  be a $2$\nbd{}square in $\CCat$. Every op\nbd{}prederivator $\sD$ induces a $2$\nbd{}square:  Leonard Guetta committed Jun 25, 2020 411  $ Leonard Guetta committed Oct 23, 2020 412 413 414 415 416  \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}  Leonard Guetta committed Jun 25, 2020 417 418 419 $ If $\sD$ has left Kan extensions, we obtain a canonical natural transformation $ Leonard Guetta committed Oct 23, 2020 420  u_!f^* \Rightarrow g^*v_!  Leonard Guetta committed Jun 25, 2020 421 $  Leonard Guetta committed Oct 23, 2020 422 423  referred to as the \emph{homological base change morphism induced by $\alpha$} and defined as the following composition:  Leonard Guetta committed Jun 25, 2020 424  $ Leonard Guetta committed Oct 23, 2020 425 426 427 428 429 430 431 432 433 434  \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}$  Leonard Guetta committed Jun 25, 2020 435   Leonard Guetta committed Dec 27, 2020 436  In particular, let $u : A \to B$ be a morphism of $\CCat$ and $b$ an object of  Leonard Guetta committed Oct 23, 2020 437 438 439 440 441 442 443 444 445 446  $B$ 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  Leonard Guetta committed Dec 27, 2020 447  $a$ an object of $A$ and $f$ an arrow of $B$, and morphisms $(a,f) \to  Leonard Guetta committed Oct 23, 2020 448 449 450 451 452  (a',f')$ 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}  Leonard Guetta committed Oct 25, 2020 453  Hence, we have a homological base change morphism:  Leonard Guetta committed Oct 23, 2020 454 455 456  $p_!\, k^* \Rightarrow b^*u_!.$  Leonard Guetta committed Oct 25, 2020 457 458  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  Leonard Guetta committed Oct 23, 2020 459 460 461 462 463 464 465 466  $\hocolim_{A/b}(X\vert_{A/b}) \rightarrow u_!(X)_b$ 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}.  Leonard Guetta committed Jun 25, 2020 467 \end{paragr}  Leonard Guetta committed Oct 23, 2020 468 469 470 471 \begin{definition}[Grothendieck] A \emph{right op-derivator} is an op\nbd{}prederivator $\sD$ such that the following axioms are satisfied: \begin{description}  Leonard Guetta committed Oct 30, 2020 472 473  \item[Der 1)] For every finite family $(A_i)_{i \in I}$ of small categories, the canonical functor  Leonard Guetta committed Oct 23, 2020 474  $ Leonard Guetta committed Jun 25, 2020 475  \sD(\amalg_{i \in I}A_i) \to \prod_{i \in I}\sD(A_i)  Leonard Guetta committed Oct 23, 2020 476 $  Leonard Guetta committed Dec 27, 2020 477  is an equivalence of categories. In particular, $\sD(\emptyset)$ is equivalent  Leonard Guetta committed Oct 23, 2020 478 479 480  to the terminal category. \item[Der 2)]\label{der2} For every small category $A$, the functor $ Leonard Guetta committed Jun 25, 2020 481  \sD(A) \rightarrow \prod_{a \in \Ob(A)}\sD(e)  Leonard Guetta committed Oct 23, 2020 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 $ 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}  Leonard Guetta committed Aug 17, 2020 509   Leonard Guetta committed Jun 25, 2020 510  \begin{example}  Leonard Guetta committed Oct 25, 2020 511  Let $\C$ be a category. The op\nbd{}prederivator represented by $\C$ always  Leonard Guetta committed Oct 23, 2020 512 513 514  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  Leonard Guetta committed Oct 25, 2020 515 516  satisfied. Hence, the op\nbd{}prederivator represented by $\C$ is a right op\nbd{}prederivator if and only if $\C$ is cocomplete.  Leonard Guetta committed Jun 25, 2020 517  \end{example}  Leonard Guetta committed Aug 17, 2020 518  \begin{remark}  Leonard Guetta committed Oct 23, 2020 519  Beware not to generalize the previous example too hastily. It is not true in  Leonard Guetta committed Oct 30, 2020 520 521  general that axiom \textbf{Der 3d} implies axiom \textbf{Der 4d}; even in the case of the homotopy op\nbd{}prederivator of a localizer.  Leonard Guetta committed Aug 17, 2020 522  \end{remark}  Leonard Guetta committed Aug 19, 2020 523 524  This motivates the following definition. \begin{definition}\label{def:cocompletelocalizer}  Leonard Guetta committed Oct 30, 2020 525 526  A localizer $(\C,\W)$ is \emph{homotopy cocomplete} if the op\nbd{}prederivator $\Ho(\C)$ is a right op-derivator.  Leonard Guetta committed Oct 23, 2020 527  \end{definition}  Leonard Guetta committed Aug 17, 2020 528  \begin{paragr}  Leonard Guetta committed Oct 23, 2020 529 530  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  Leonard Guetta committed Oct 25, 2020 531  op\nbd{}prederivator has right Kan extensions and that they are computed  Leonard Guetta committed Oct 30, 2020 532 533 534 535  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  Leonard Guetta committed Aug 17, 2020 536 537 538 539  \begin{align*} \CCat &\to \CCAT \\ A &\mapsto (\sD(A^{\op}))^{\op} \end{align*}  Leonard Guetta committed Oct 30, 2020 540 541 542  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.  Leonard Guetta committed Aug 17, 2020 543  \end{paragr}  Leonard Guetta committed Jun 25, 2020 544  \section{Morphisms of op-derivators, preservation of homotopy colimits}  Leonard Guetta committed Jan 20, 2021 545  \sectionmark{Morphisms of op-derivators}  Leonard Guetta committed Oct 23, 2020 546 547 548  We refer to \cite{leinster1998basic} for the precise definitions of pseudo-natural transformation (called strong transformation there) and modification.  Leonard Guetta committed Jun 25, 2020 549  \begin{paragr}  Leonard Guetta committed Oct 25, 2020 550  Let $\sD$ and $\sD'$ be two op\nbd{}prederivators. A \emph{morphism of  Leonard Guetta committed Oct 30, 2020 551 552  op\nbd{}prederivators} $F : \sD \to \sD'$ is a pseudo-natural transformation from $\sD$ to $\sD'$. This means that $F$ consists of:  Leonard Guetta committed Jun 25, 2020 553 554  \begin{itemize}[label=-] \item a functor $F_A : \sD(A) \to \sD'(A)$ for every small category $A$,  Leonard Guetta committed Oct 23, 2020 555 556  \item an isomorphism of functors $F_u: F_A u^* \overset{\sim}{\Rightarrow} u^* F_B$,  Leonard Guetta committed Jun 25, 2020 557  $ Leonard Guetta committed Oct 23, 2020 558 559 560 561 562  \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}  Leonard Guetta committed Jun 25, 2020 563 564 565 $ for every $u : A \to B$ in $\CCat$. \end{itemize}  Leonard Guetta committed Oct 23, 2020 566 567  These data being compatible with compositions and units. The morphism is \emph{strict} when $F_u$ is an identity for every $u : A \to B$.  Leonard Guetta committed Jun 25, 2020 568   Leonard Guetta committed Oct 23, 2020 569  Let $F : \sD \to \sD'$ and $G : \sD \to \sD'$ be morphisms of  Leonard Guetta committed Oct 30, 2020 570 571 572 573 574  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.  Leonard Guetta committed Jun 25, 2020 575   Leonard Guetta committed Oct 30, 2020 576 577 578  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.  Leonard Guetta committed Jun 25, 2020 579 580 581  \end{paragr} \begin{example}  Leonard Guetta committed Dec 27, 2020 582  Let $F : \C \to \C'$ be a functor. It induces a strict morphism at the level  Leonard Guetta committed Oct 30, 2020 583 584 585 586  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.  Leonard Guetta committed Jun 25, 2020 587 588  \end{example} \begin{example}  Leonard Guetta committed Oct 23, 2020 589 590  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  Leonard Guetta committed Oct 25, 2020 591  equivalences and the universal property of the localization yields a  Leonard Guetta committed Oct 23, 2020 592  functor $\overline{F}_A : \ho(\C(A)) \to \ho(\C'(A)).$ This defines a  Leonard Guetta committed Oct 25, 2020 593  strict morphism of op\nbd{}prederivators:  Leonard Guetta committed Jun 25, 2020 594  $ Leonard Guetta committed Oct 23, 2020 595  \overline{F} : \Ho(\C) \to \Ho(\C').  Leonard Guetta committed Jun 25, 2020 596 $  Leonard Guetta committed Oct 25, 2020 597  Similarly, every $2$\nbd{}morphism of localizers  Leonard Guetta committed Aug 17, 2020 598  $ Leonard Guetta committed Oct 23, 2020 599 600 601  \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]  Leonard Guetta committed Aug 17, 2020 602 603  \end{tikzcd}$  Leonard Guetta committed Oct 24, 2020 604 605  induces a $2$\nbd{}morphism $\overline{\alpha} : \overline{F} \Rightarrow \overline{G}$. Altogether, we have defined a $2$\nbd{}functor  Leonard Guetta committed Aug 17, 2020 606 607  \begin{align*} \Loc &\to \PPder\\  Leonard Guetta committed Aug 18, 2020 608 609  (\C,\W) &\mapsto \Ho(\C), \end{align*}  Leonard Guetta committed Oct 24, 2020 610  where $\Loc$ is the $2$\nbd{}category of localizers.  Leonard Guetta committed Aug 18, 2020 611  \end{example}  Leonard Guetta committed Sep 17, 2020 612  \begin{paragr}\label{paragr:canmorphismcolimit}  Leonard Guetta committed Oct 30, 2020 613  Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions  Leonard Guetta committed Dec 27, 2020 614  and let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators. For every $u :  Leonard Guetta committed Oct 30, 2020 615  A \to B$, there is a canonical natural transformation  Leonard Guetta committed Aug 18, 2020 616  $ Leonard Guetta committed Oct 30, 2020 617  u_!\, F_A \Rightarrow F_B\, u_!  Leonard Guetta committed Aug 18, 2020 618 619 620 $ defined as $ Leonard Guetta committed Oct 23, 2020 621 622 623 624 625 626 627 628  \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}$  Leonard Guetta committed Oct 30, 2020 629 630 631  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  Leonard Guetta committed Oct 23, 2020 632 633 634  $\hocolim_{A}(F_A(X))\to F_e(\hocolim_A(X)).$  Leonard Guetta committed Aug 18, 2020 635 636  \end{paragr} \begin{definition}\label{def:cocontinuous}  Leonard Guetta committed Dec 27, 2020 637  Let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators and suppose that  Leonard Guetta committed Oct 30, 2020 638  $\sD$ and $\sD'$ both admit left Kan extensions. We say that $F$ is  Leonard Guetta committed Dec 27, 2020 639  \emph{cocontinuous}\footnote{Some authors also say \emph{left exact}.} if for every $u: A \to B$, the  Leonard Guetta committed Oct 23, 2020 640  canonical morphism  Leonard Guetta committed Aug 18, 2020 641  $ Leonard Guetta committed Oct 30, 2020 642  u_! \, F_A \Rightarrow F_B \, u_!  Leonard Guetta committed Aug 18, 2020 643 644 645 646 $ is an isomorphism. \end{definition} \begin{remark}  Leonard Guetta committed Oct 30, 2020 647 648 649  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  Leonard Guetta committed Oct 23, 2020 650  extensions.  Leonard Guetta committed Aug 18, 2020 651 652  \end{remark} \begin{example}  Leonard Guetta committed Oct 23, 2020 653 654  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  Leonard Guetta committed Oct 30, 2020 655 656  op\nbd{}prederivators is cocontinuous if and only if $F$ is cocontinuous in the usual sense.  Leonard Guetta committed Jun 25, 2020 657  \end{example}  Leonard Guetta committed Sep 02, 2020 658  \begin{paragr}\label{paragr:prederequivadjun}  Leonard Guetta committed Oct 30, 2020 659 660  As in any $2$\nbd{}category, the notions of equivalence and adjunction make sense in $\PPder$. Precisely, we have that:  Leonard Guetta committed Aug 18, 2020 661  \begin{itemize}  Leonard Guetta committed Oct 30, 2020 662 663 664  \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  Leonard Guetta committed Oct 23, 2020 665  $\mathrm{id}_{\sD}$; the morphism $G$ is a \emph{quasi-inverse} of $F$.  Leonard Guetta committed Oct 30, 2020 666 667 668 669 670  \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.  Leonard Guetta committed Aug 18, 2020 671 672  \end{itemize} \end{paragr}  Leonard Guetta committed Oct 30, 2020 673 674  The following three lemmas are easy $2$\nbd{}categorical routine and are left to the reader.  Leonard Guetta committed Aug 18, 2020 675  \begin{lemma}\label{lemma:dereq}  Leonard Guetta committed Oct 25, 2020 676  Let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators. If $F$ is an  Leonard Guetta committed Oct 23, 2020 677 678  equivalence then $\sD$ is a right op-derivator (resp.\ left op-derivator, resp.\ op-derivator) if and only if $\sD'$ is one.  Leonard Guetta committed Aug 18, 2020 679  \end{lemma}  Leonard Guetta committed Sep 05, 2020 680  \begin{lemma}\label{lemma:eqisadj}  Leonard Guetta committed Oct 23, 2020 681 682 683  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}  Leonard Guetta committed Sep 05, 2020 684  \begin{lemma}\label{lemma:ladjcocontinuous}  Leonard Guetta committed Oct 30, 2020 685 686 687  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.  Leonard Guetta committed Aug 18, 2020 688  \end{lemma}  Leonard Guetta committed Oct 23, 2020 689  We end this section with a generalization of the notion of localization in the  Leonard Guetta committed Oct 25, 2020 690  context of op\nbd{}prederivators.  Leonard Guetta committed Oct 23, 2020 691  \begin{paragr}  Leonard Guetta committed Jun 25, 2020 692 693  Let $(\C,\W)$ be a localizer. For every small category $A$, let $ Leonard Guetta committed Oct 23, 2020 694  \gamma_A : \C(A) \to \ho(\C(A))  Leonard Guetta committed Jun 25, 2020 695 $  Leonard Guetta committed Oct 30, 2020 696 697  be the localization functor. The correspondence $A \mapsto \gamma_A$ is natural in $A$ and defines a strict morphism of  Leonard Guetta committed Oct 25, 2020 698  op\nbd{}prederivators  Leonard Guetta committed Jun 25, 2020 699  $ Leonard Guetta committed Oct 23, 2020 700  \gamma : \C \to \Ho(\C).  Leonard Guetta committed Jun 25, 2020 701 702 703 $ \end{paragr} \begin{definition}\label{def:strnglyder}  Leonard Guetta committed Oct 23, 2020 704 705  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  Leonard Guetta committed Oct 25, 2020 706  morphism of op\nbd{}prederivators  Leonard Guetta committed Jun 25, 2020 707  $ Leonard Guetta committed Oct 23, 2020 708  \LL F : \Ho(\C) \to \Ho(\C')  Leonard Guetta committed Jun 25, 2020 709 $  Leonard Guetta committed Oct 25, 2020 710  and a $2$\nbd{}morphism of op\nbd{}prederivators  Leonard Guetta committed Jun 25, 2020 711  $ Leonard Guetta committed Oct 23, 2020 712 713 714 715 716 717 718 719 720  \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  Leonard Guetta committed Oct 30, 2020 721 722 723 724  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.  Leonard Guetta committed Jun 25, 2020 725  \end{definition}  Leonard Guetta committed Aug 18, 2020 726  \begin{example}  Leonard Guetta committed Oct 23, 2020 727 728 729 730  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 $\overline{F} \simeq \LL F \simeq \RR G.$  Leonard Guetta committed Aug 18, 2020 731  \end{example}  Leonard Guetta committed Oct 23, 2020 732 733  Gonzalez' criterion (Proposition \ref{prop:gonz}) admits the following generalization.  Leonard Guetta committed Aug 20, 2020 734  \begin{proposition}\label{prop:gonzalezcritder}  Leonard Guetta committed Aug 18, 2020 735 736  Let $(\C,\W)$ and $(\C',\W')$ be two localizers and $ Leonard Guetta committed Oct 23, 2020 737 738  \begin{tikzcd} F : \C \ar[r,shift left] & \C' : G \ar[l,shift left]  Leonard Guetta committed Aug 18, 2020 739 740  \end{tikzcd}$  Leonard Guetta committed Oct 23, 2020 741 742 743 744 745  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  Leonard Guetta committed Aug 18, 2020 746  $ Leonard Guetta committed Oct 23, 2020 747  \LL F \simeq F'.  Leonard Guetta committed Aug 18, 2020 748 749 $ \end{proposition}  Leonard Guetta committed Jun 25, 2020 750  \begin{proof}  Leonard Guetta committed Oct 30, 2020 751 752 753  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.  Leonard Guetta committed Oct 23, 2020 754  Proposition \ref{prop:gonz} gives us that for every small category $A$, the  Leonard Guetta committed Oct 30, 2020 755  functor $F_A$ is absolutely totally left derivable with $(F'_A,\alpha_A)$  Leonard Guetta committed Oct 23, 2020 756 757  its total left derived functor. This means exactly that $F'$ is strongly left derivable and $(F',\alpha)$ is the left derived morphism of  Leonard Guetta committed Oct 25, 2020 758  op\nbd{}prederivators of $F$.  Leonard Guetta committed Jun 25, 2020 759  \end{proof}  Leonard Guetta committed Aug 18, 2020 760  \section{Homotopy cocartesian squares}  Leonard Guetta committed Jun 25, 2020 761  \begin{paragr}  Leonard Guetta committed Oct 25, 2020 762  Let $\Delta_1$ be the ordered set $\{0 <1\}$ seen as category. We use the  Leonard Guetta committed Oct 30, 2020 763 764  notation $\square$ for the category $\Delta_1\times \Delta_1$, which can be pictured as the \emph{commutative} square  Leonard Guetta committed Jun 25, 2020 765  $ Leonard Guetta committed Oct 23, 2020 766 767 768  \begin{tikzcd} (0,0) \ar[r] \ar[d] & (0,1)\ar[d] \\ (1,0) \ar[r] & (1,1)  Leonard Guetta committed Aug 18, 2020 769  \end{tikzcd}  Leonard Guetta committed Jun 25, 2020 770 $  Leonard Guetta committed Oct 23, 2020 771 772  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  Leonard Guetta committed Jun 25, 2020 773  $ Leonard Guetta committed Oct 23, 2020 774 775 776  \begin{tikzcd} (0,0) \ar[d] \ar[r] & (0,1) \\ (1,0) &.  Leonard Guetta committed Oct 14, 2020 777  \end{tikzcd}  Leonard Guetta committed Jun 25, 2020 778 $  Leonard Guetta committed Oct 23, 2020 779 780  Finally, we write $i_{\ulcorner} : \ulcorner \to \square$ for the canonical inclusion functor.  Leonard Guetta committed Aug 18, 2020 781  \end{paragr}  Leonard Guetta committed Aug 18, 2020 782  \begin{definition}\label{def:cocartesiansquare}  Leonard Guetta committed Oct 25, 2020 783  Let $\sD$ be an op\nbd{}prederivator. An object $X$ of $\sD(\square)$ is  Leonard Guetta committed Dec 27, 2020 784  \emph{cocartesian} if for every object $Y$ of $\sD(\square)$, the canonical  Leonard Guetta committed Oct 23, 2020 785  map  Leonard Guetta committed Aug 18, 2020 786  $ Leonard Guetta committed Oct 23, 2020 787 788  \Hom_{\sD(\square)}(X,Y) \to \Hom_{\sD(\ulcorner)}(i_{\ulcorner}^*(X),i_{\ulcorner}^*(Y))  Leonard Guetta committed Aug 18, 2020 789 $  Leonard Guetta committed Oct 23, 2020 790 791  induced by the functor $i_{\ulcorner}^* : \sD(\square) \to \sD(\ulcorner)$, is a bijection.  Leonard Guetta committed Jun 25, 2020 792  \end{definition}  Leonard Guetta committed Aug 18, 2020 793  \begin{example}  Leonard Guetta committed Oct 23, 2020 794 795 796  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.  Leonard Guetta committed Aug 18, 2020 797  \end{example}  Leonard Guetta committed Oct 23, 2020 798 799 800 801 802  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).  Leonard Guetta committed Aug 18, 2020 803  \begin{definition}\label{def:hmtpycocartesiansquare}  Leonard Guetta committed Oct 23, 2020 804 805 806  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}.  Leonard Guetta committed Aug 18, 2020 807 808  \end{definition} \begin{paragr}  Leonard Guetta committed Oct 25, 2020 809  Let $\sD$ be an op\nbd{}prederivator. The object $(1,1)$ of $\square$ can be  Leonard Guetta committed Oct 23, 2020 810  considered as a morphism of $\Cat$  Leonard Guetta committed Aug 19, 2020 811  $ Leonard Guetta committed Oct 23, 2020 812  (1,1) : e \to \square  Leonard Guetta committed Aug 19, 2020 813 $  Leonard Guetta committed Oct 23, 2020 814 815  and thus induces a functor $(1,1)^* : \sD(\square)\to \sD(e)$. For an object $X$ of $\sD(\square)$, we use the notation  Leonard Guetta committed Aug 19, 2020 816  $ Leonard Guetta committed Oct 23, 2020 817  X_{(1,1)} := (1,1)^*(X).  Leonard Guetta committed Aug 19, 2020 818 $  Leonard Guetta committed Oct 23, 2020 819  Now, since $(1,1)$ is the terminal object of $\square$, we have a canonical  Leonard Guetta committed Oct 24, 2020 820  $2$\nbd{}triangle  Leonard Guetta committed Aug 19, 2020 821  $ Leonard Guetta committed Oct 23, 2020 822 823 824 825  \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}  Leonard Guetta committed Aug 19, 2020 826 $  Leonard Guetta committed Oct 23, 2020 827  where we wrote $p$ instead of $p_{\ulcorner}$ for short. Hence, we have a  Leonard Guetta committed Oct 24, 2020 828  $2$\nbd{}triangle  Leonard Guetta committed Aug 19, 2020 829  $ Leonard Guetta committed Oct 23, 2020 830 831 832 833  \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}  Leonard Guetta committed Aug 19, 2020 834 $  Leonard Guetta committed Oct 23, 2020 835 836 837  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  Leonard Guetta committed Aug 19, 2020 838  \[  Leonard Guetta committed Oct 23, 2020 839  p_{!}(i_{\ulcorner}^*(X)) \to p_{!} p^* (X_{(1,1)}) \to X_{(1,1)},