Commit 1466d455 by Leonard Guetta

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parent 2a1fccac
 ... ... @@ -51,10 +51,10 @@ is poorly behaved. For example, \fi $\gamma(X)=X$ for any object $X$ of $\C$. for every object $X$ of $\C$. The class of arrows $\W$ is said to be \emph{saturated} when we have the property property: $f \in \W \text{ if and only if } \gamma(f) \text{ is an isomorphism. }$ ... ... @@ -67,14 +67,14 @@ For later reference, we put here the following definition. $\begin{tikzcd} X \ar[r] \ar[d,"f"] & X' \ar[d,"f'"] \\ Y \ar[r] & Y' \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] Y \ar[r] & Y', \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd}$ 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 $F(\W) \subseteq$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 $... ... @@ -88,7 +88,7 @@ For later reference, we put here the following definition. \end{tikzcd}$ is commutative. Let$G : (\C,\W) \to (\C',\W')$be another morphism of localizers. A \emph{2-morphism of localizers} from$F$to$G$is simply a 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 canonical natural transformation $... ... @@ -96,7 +96,7 @@ For later reference, we put here the following definition. \ar[r,bend right,"\overline{G}"',""{name=B,above}] & \ho(\C') \ar[from=A,to=B,Rightarrow,"\overline{\alpha}"]\end{tikzcd}$ such that the 2-diagram 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"] \\ ... ... @@ -108,7 +108,7 @@ For later reference, we put here the following definition. is commutative in an obvious sense. \end{paragr} \begin{remark}\label{remark:localizedfunctorobjects} Since we always consider that for any localizer (\C,\W) the categories \C Since we always consider that for every localizer (\C,\W) the categories \C and \ho(\C) have the same objects and the localization functor is the identity on objects, it follows that for a morphism of localizer {F : (\C,\W) \to (\C',\W')}, we tautologically have ... ... @@ -143,7 +143,7 @@ For later reference, we put here the following definition. F}. Often we will abusively discard \alpha and simply refer to \LL F as the total left derived functor of F. The notion of total right derivable functor is defined dually and denoted by 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} ... ... @@ -180,7 +180,7 @@ we shall use in the sequel. 3.1]{gonzalez2012derivability}}]\label{prop:gonz} Let (\C,\W) and (\C',\W') be two localizers and \[\begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] : G \end{tikzcd}$ an adjunction. If$G$is absolutely totally right G \end{tikzcd}\] 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] : ... ... @@ -192,10 +192,10 @@ we shall use in the sequel. % faudrait vérifier que je n'ai pas dit de bêtises en le reformulant.} \section{(op)Derivators and homotopy colimits} \begin{notation}We denote by \CCat the 2-category of small categories and \CCAT the 2-category of big categories. For a 2-category \underline{A}, the 2-category obtained from \underline{A} by switching the source and targets of 1-cells is denoted by \underline{A}^{op}. \begin{notation}We denote by \CCat the 2\nbd{}category of small categories and \CCAT the 2\nbd{}category of big categories. For a 2\nbd{}category \underline{A}, the 2\nbd{}category obtained from \underline{A} by switching the source and targets of 1-cells is denoted by \underline{A}^{\op}. The terminal category, i.e.\ the category with only one object and no non-trivial arrows, is canonically denoted by e. For a (small) category A, ... ... @@ -205,7 +205,7 @@ we shall use in the sequel.$ \end{notation} \begin{definition} An \emph{op-prederivator} is a (strict)$2$-functor An \emph{op-prederivator} is a (strict)$2$\nbd{}functor $\sD : \CCat^{op} \to \CCAT.$ More explicitly, an op-prederivator consists of the data of: \begin{itemize}[label=-] ... ... @@ -262,14 +262,14 @@ we shall use in the sequel. Note that some authors call \emph{prederivator} what we have called \emph{op-prederivator}. The terminology we chose in the above definition is compatible with the original one of Grothendieck, who called \emph{prederivator} a$2$-functor from$\CCat$to$\CCAT$that is contravariant at the level of$1$-cells \emph{and} at the level of$2$-cells. \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$2$-functorial in an obvious sense and thus defines an op-prederivator$2\nbd{}functorial in an obvious sense and thus defines an op-prederivator \begin{align*} \C : \CCat^{op} &\to \CCAT \\ A &\mapsto \C(A) ... ... @@ -286,10 +286,10 @@ We now turn to the most important way of obtaining op-prederivators. \begin{paragr}\label{paragr:homder} Let(\C,\W)$be a localizer. For every small category$A$, there is a localizer$(\C(A),\W_A)$, where$\W_A$the class of \emph{pointwise weak equivalences}, i.e. arrows$\alpha : d \to d'$of$\C(A)$such that equivalences}, i.e.\ arrows$\alpha : d \to d'$of$\C(A)$such that$\alpha_a : d(a) \to d'(a)$belongs to$\W$for every$a \in \Ob(A)$. The correspondence$A \mapsto (\C(A),\W_A)$is$2$-functorial in that every$u 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) ... ... @@ -297,7 +297,7 @@ We now turn to the most important way of obtaining op-prederivators. 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-morphism of localizers a 2\nbd{}morphism of localizers \[ \begin{tikzcd} (\C(B),\W_B) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, ... ... @@ -305,10 +305,10 @@ We now turn to the most important way of obtaining op-prederivators. \ar[from=A,to=B,Rightarrow,"\alpha^*"] \end{tikzcd}$ (This last property is trivial since a $2$-morphism of localizers is simply a (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 localization, we have for every $u : A \to B$ an induced functor, which we still denote $u^*$, universal property of the localization, we have for every $u : A \to B$, an induced functor, which we still denote by $u^*$, $u^* : \ho(\C(B)) \to \ho(\C(A))$ ... ... @@ -349,7 +349,7 @@ We now turn to the most important way of obtaining op-prederivators. extensions if and only if the category $\C$ has left Kan extensions along any morphism $u : A \to B$ of $\Cat$ in the usual sense. By a standard categorical argument, this means that the op-prederivator represented by $\C$ has left Kan extensions if and only if $\C$ is cocomplete. Note that for any small category 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) ... ... @@ -362,15 +362,15 @@ We now turn to the most important way of obtaining op-prederivators.$ \end{example} \begin{paragr} A localizer $(\C,\W)$ has \emph{homotopy left Kan extensions} when the We say that a localizer $(\C,\W)$ has \emph{homotopy left Kan extensions} when the homotopy op-prederivator of $(\C,\W)$ has left Kan extensions. In this case, for any small category $A$, the \emph{homotopy colimit functor of $A$-shaped 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$ if $\ho(\C(A))$ (which is nothing but a diagram $X : A \to \C$, only seen up to weak equivalence''), the object of $\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)$ ... ... @@ -402,8 +402,8 @@ We now turn to the most important way of obtaining op-prederivators. C \ar[r,"g"'] & D \ar[from=1-2,to=2-1,Rightarrow,"\alpha"] \end{tikzcd} \] be a $2$-square in $\CCat$. For any op-prederivator $\sD$, we have an induced $2$-square: be a $2$\nbd{}square in $\CCat$. For any op-prederivator $\sD$, we have an induced $2$\nbd{}square: $\begin{tikzcd} \sD(A) & \sD(B) \ar[l,"f^*"'] \\ ... ... @@ -562,13 +562,13 @@ We now turn to the most important way of obtaining op-prederivators. \emph{strict} when F_u is an identity for every u : A \to B. Let F : \sD \to \sD' and G : \sD \to \sD' be morphisms of op-prederivators. A \emph{2-morphism \phi : F \to G} is a modification op-prederivators. A \emph{2\nbd{}morphism \phi : F \to 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-category of op-prederivators, morphisms of op-prederivators and 2-morphisms of op-prederivators. We denote by \PPder the 2\nbd{}category of op-prederivators, morphisms of op-prederivators and 2\nbd{}morphisms of op-prederivators. \end{paragr} \begin{example} ... ... @@ -576,7 +576,7 @@ We now turn to the most important way of obtaining op-prederivators. denoted by F, at the level of op-prederivators, where for every small category A, the functor F_A : \C(A) \to \C'(A) is induced by post-composition. Similarly, any natural transformation induces a 2-morphism at the level of represented op-prederivators. 2\nbd{}morphism at the level of represented op-prederivators. \end{example} \begin{example} Let F : (\C,\W) \to (\C',\W') be a morphism of localizers. For every small ... ... @@ -587,20 +587,20 @@ We now turn to the most important way of obtaining op-prederivators. \[ \overline{F} : \Ho(\C) \to \Ho(\C').$ Similarly, any $2$-morphism of localizers Similarly, any $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$-morphism $\overline{\alpha} : \overline{F} \Rightarrow \overline{G}$. Altogether, we have defined a $2$-functor 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$-category of localizers. where $\Loc$ is the $2$\nbd{}category of localizers. \end{example} \begin{paragr}\label{paragr:canmorphismcolimit} Let $\sD$ and $\sD'$ be op-prederivators that admit left Kan extensions and ... ... @@ -649,7 +649,7 @@ We now turn to the most important way of obtaining op-prederivators. usual sense. \end{example} \begin{paragr}\label{paragr:prederequivadjun} As in any $2$-category, the notions of equivalence and adjunction make sense 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-prederivators $F : \sD \to \sD'$ is an equivalence ... ... @@ -658,12 +658,12 @@ We now turn to the most important way of obtaining op-prederivators. $\mathrm{id}_{\sD}$; the morphism $G$ is a \emph{quasi-inverse} of $F$. \item[-] A morphism of op-prederivators $F : \sD \to \sD'$ is left adjoint to $G : \sD' \to \sD$ (and $G$ is right adjoint to $F$) when there exist $2$-morphisms $\eta : \mathrm{id}_{\sD'} \Rightarrow GF$ and $\epsilon :$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$-categorical routine and are left to 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-prederivators. If$F$is an ... ... @@ -699,7 +699,7 @@ We now turn to the most important way of obtaining op-prederivators. $\LL F : \Ho(\C) \to \Ho(\C')$ and a$2$-morphism of op-prederivators and a$2$\nbd{}morphism of op-prederivators $\begin{tikzcd} \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\ ... ... @@ -740,7 +740,7 @@ We now turn to the most important way of obtaining op-prederivators.$ \end{proposition} \begin{proof} Let$\alpha : F' \circ \gamma \Rightarrow F\circ \gamma$be the$2$-morphism Let$\alpha : F' \circ \gamma \Rightarrow F\circ \gamma$be the$2$\nbd{}morphism of op-prederivators defined \emph{mutatis mutandis} as in \ref{paragr:prelimgonzalez} but at the level of op-prederivators. Proposition \ref{prop:gonz} gives us that for every small category$A$, the ... ... @@ -809,7 +809,7 @@ We now turn to the most important way of obtaining op-prederivators. X_{(1,1)} := (1,1)^*(X). \] Now, since$(1,1)$is the terminal object of$\square$, we have a canonical$2$-triangle$2$\nbd{}triangle $\begin{tikzcd} \ulcorner \ar[rr,"i_{\ulcorner}",""{name=A,below}] \ar[rd,"p"'] && \square\\ ... ... @@ -817,7 +817,7 @@ We now turn to the most important way of obtaining op-prederivators. \end{tikzcd}$ where we wrote$p$instead of$p_{\ulcorner}$for short. Hence, we have a$2$-triangle$2$\nbd{}triangle $\begin{tikzcd} \sD(\ulcorner) && \sD(\square) \ar[ll,"i_{\ulcorner}^*"',""{name=A,below}] \ar[dl,"{(1,1)}^*"] \\ ... ... No preview for this file type  ... ... @@ -2161,7 +2161,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an \omega-functo \[ \wt{F}_a : \G[\Sigma^C]_a \to \G[\Sigma^D]_{F(a)}$ for any$a \in C_{n}$. for every$a \in C_{n}$. \end{paragr} \begin{lemma}\label{lemmafaithful} With the notations of the above paragraph, the map $... ... @@ -2213,7 +2213,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an \omega-functo \mu =(v_1,v_2,e,e') : v \to v'$ be an elementary move. Since, by definition, $\wt{F}(u)= v = v_1ev_2$ $\wt{F}(u)= v = v_1ev_2,$$u$is necessarily of the form $u=u_1\overline{e}u_2 ... ... @@ -2248,12 +2248,12 @@ Recall from Proposition \ref{prop:conduchepractical} that for an \omega-functo \wt{F}(\lambda)=\mu.$ All that is left to prove now is ythe existence of$\overline{e'}$with the desired properties. All that is left to prove now is the existence of$\overline{e'}$with the desired properties. \begin{description} \item[First case:] The word$e$is of the form $((x\fcomp_k y )\fcomp_k z)$ and$e'$is of the form and the word$e'$is of the form $(x\fcomp_k (y \fcomp_k z))$ with$x,y,z \in \T[\Sigma^D]$. The word$\overline{e}$is then necessarily of the form ... ... @@ -2278,7 +2278,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an$\omega$-functo $(x\fcomp_k(\ii_{\1^{n-1}_z}))$ and$e'$is of the form and the word$e'$is of the form $x$ ... ... @@ -2296,7 +2296,7 @@ Recall from Proposition \ref{prop:conduchepractical} that for an$\omega$-functo $\wt{F}(y)=\1^{n-1}_z.$ Then we set Then, we set $\overline{e'}:=x.$ ... ... @@ -2312,7 +2312,7 @@ If$k 0$,$\Sigma^D \subseteq D_{n}$and$\Sigma^C = F^{-1}(\Sigma^D)$. If$\tau^s_{\leq n}(F)$is a discrete Conduché$n$\nbd{}functor, then for every$a \in C_{n}$Let$F : C \to D$be an$\omega$-functor,$n>0$,$\Sigma^D \subseteq D_{n}$and$\Sigma^C := F^{-1}(\Sigma^D)$. If$\tau^s_{\leq n}(F)$is a discrete Conduché$n$\nbd{}functor, then for every$a \in C_{n}$$\wt{F}_a : \G[\Sigma^C]_a\to \G[\Sigma^D]_{F(a)}$ ... ... @@ -2417,13 +2417,13 @@ If$k 0$. \begin{proof}The case$n=0$is trivial. We now suppose that$n>0$. From Lemma \ref{lemma:isomorphismgraphs} we have that for every$a \in C_n$, the map $\wt{F}_a : \G[\Sigma^C]_a \to \G[\Sigma^D]_{F(a)} ... ... @@ -2437,7 +2437,7 @@ Putting all the pieces together, we finally have the awaited proof. \item In the case that D is free, it follows immediately from the first part of the Proposition \ref{prop:conduchenbasis} that C is free. \item In the case that C is free and F_n : C_n \to D_n is surjective for every n \in \mathbb{N}, let us write \Sigma_n^C for the n\nbd{}basis of C. It follows from Proposition \ref{prop:uniquebasis} and Lemma \ref{lemma:conducheindecomposable} that \[ F^{-1}(F(\Sigma^C_n)=\Sigma_n^C. F^{-1}(F(\Sigma^C_n))=\Sigma_n^C.$ Hence, we can apply the second part of Proposition \ref{prop:conduchenbasis} and$C\$ is free. \end{enumerate} ... ...
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