Commit 6b0c8526 authored by Leonard Guetta's avatar Leonard Guetta
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Edited typos. Is it over ?

parent c4d240c8
......@@ -1974,16 +1974,21 @@ homotopy type of the torus.
$1$\nbd{}category, it is \good{} (Theorem \ref{thm:categoriesaregood}), which
means that the right vertical arrow is an isomorphism. The $1$\nbd{}category
$B^1(\mathbb{N}\times \mathbb{N})$ is not free but since it has the homotopy
type of the torus, we have $H^{\sing}_k(B^1(\mathbb{N}\times \mathbb{N}))=0=H_k^{\pol}(\mathbb{N}\times \mathbb{N}))$
type of the torus, we have $H^{\sing}_k(B^1(\mathbb{N}\times \mathbb{N}))=0=H_k^{\pol}(B^1(\mathbb{N}\times \mathbb{N}))$
for $k\geq 2$ and it follows then from Corollary \ref{cor:polhmlgycofibrant}
and Paragraph \ref{paragr:polhmlgylowdimension} that the map $\sH^{\pol}(F)$
may be identified with the image in $\ho(\Ch)$ of the map
and Paragraph \ref{paragr:polhmlgylowdimension} that the map canonical map
\[
\lambda(F) : \lambda(P) \to \lambda(B^1(\mathbb{N}\times\mathbb{N})).
\alpha^{\pol}_{B^{1}(\mathbb{N}\times\mathbb{N})} :
\sH^{\pol}(B^{1}(\mathbb{N}\times\mathbb{N})) \to \lambda(B^{1}(\mathbb{N}\times\mathbb{N}))
\]
It is straightforward to check that this last map is a quasi-isomorphism,
which implies by a 2-out-of-3 property that $\pi_P : \sH^{\pol}(P) \to
\sH^{\sing}(P)$ is an isomorphism. This means by definition that $P$ is
is a quasi-isomorphism. Since $P$ is free, it follows that the map $\sH^{\pol}(F)$
can be identified with the image in $\ho(\Ch)$ of the map
\[
\lambda(F) : \lambda(P) \to \lambda(B^1(\mathbb{N}\times\mathbb{N})),
\]
which is easily checked to be a quasi-isomorphism.
By a 2-out-of-3 property, we deduce that $\pi_P : \sH^{\pol}(P) \to
\sH^{\sing}(P)$ is an isomorphism, which means by definition that $P$ is
\good{}.
\end{paragr}
% \begin{paragr}
......
......@@ -135,7 +135,7 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re
\sD_n \ar[r,"j_n^-"'] & \sS_{n}.
\end{tikzcd}
\]
is cartesian all of the four morphisms are monomorphisms. Since the
is cartesian and all four morphisms are monomorphisms. Since the
functor $\Hom_{\oo\Cat}(\Or_k,-)$ preserves limits, the square
\eqref{squarenervesphere} is a cartesian square of $\Set$ all of
whose four morphisms are monomorphisms. Hence, in order to prove
......@@ -160,7 +160,21 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re
\begin{description}
\item[Case $k<n$:] Since every generating cell of $\gamma$ of $\Or_k$ is of dimension non-greater than $k$, the cell $\varphi(\gamma)$ is of dimension strictly lower than $n$. Since all cells of dimension strictly lower than $n$ are both in the image of $j^+_n$ and in the image of $j^-_n$, $\varphi$ obviously factors through $j^+_n$ (and $j^+_n$).
\item[Case $k=n$:] The image of $\alpha_n$ is either a non-trivial $n$\nbd{}cell of $\sS_n$ or a unit on a strictly lower dimensional cell. In the second situation, everything works like the case $k<n$. Now suppose for example that $\varphi(\alpha_n)$ is $h^+_n$, which is in the image of $j^+_n$. Since all of the other generating cells of $\Or_n$ are of dimension strictly lower than $n$, their images by $\varphi$ are also of dimension strictly lower than $n$ and hence, are all contained in the image of $j^+_n$. Altogether this proves that $\varphi$ factors through $j^+_n$. The case where $\varphi(\alpha_n)=h^-_n$ is symmetric.
\item[Case $k>n$:] Since $\sS_n$ is an $n$\nbd{}category, the image of $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If $x$ is a unit on a cell whose dimension is strictly lower than $n$, then everything works like in the case $k<n$. If not, this means that $x$ is a non-trivial $n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let $\gamma$ be a generator of $\Or_k$ of dimension $k-1$. We have $\varphi(\gamma)=\1^{k-1}_y$ with $y$ which is either a unit on a cell of dimension strictly lower than $n$, or a non-degenerate $n$\nbd{}cell of $\sS_n$ (if $k-1=n$, recall the convention that $\1^{k-1}_y=y$). In the first situation, $y$ is in the image of $j^+_n$ as in the case $k<n$, and thus, so is $\1^{k-1}_y$. In the second situation, this means \emph{a priori} that either $y=h_n^+$ or $y=h_n^-$. But we know that $\gamma$ is part of a composition that is equal to either the source or the target of $\alpha_k$ (see \ref{paragr:orientals}) and thus, $f(\gamma)$ is part of a composition that is equal to either the source or the target of $x=h^+_n$. Since no composition involving $h^-_n$ can be equal to $h^+_n$ (one could invoke the function introduced in \ref{prop:countingfunction}), this implies that $y=h_n^+$ and hence, $f(\gamma)$ is in the image of $j^+_n$. This goes for all generating cells of dimension $k-1$ of $\Or_k$ and we can recursively apply the same reasoning for generating cells of dimension $k-2$, then $k-3$ and so forth. Altogether, this proves that $\varphi$ factorizes through $j^+_n$. The case where $x=h^-_n$ and $\varphi$ factorizes through $j^-_n$ is symmetric.
\item[Case $k>n$:] Since $\sS_n$ is an $n$\nbd{}category, the image of
$\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$
a cell of $\sS_n$ of dimension non-greater than $n$. If $x$ is a unit on a
cell whose dimension is strictly lower than $n$, then everything works like
in the case $k<n$. If not, this means that $x$ is a non-trivial
$n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let
$\gamma$ be a generator of $\Or_k$ of dimension $k-1$. We have
$\varphi(\gamma)=\1^{k-1}_y$ with $y$ which is either a unit on a cell of
dimension strictly lower than $n$, or a non-degenerate $n$\nbd{}cell of
$\sS_n$ (if $k-1=n$, we use the convention that $\1^{k-1}_y=y$). In the
first situation, $y$ is in the image of $j^+_n$ as in the case $k<n$, and
thus, so is $\1^{k-1}_y$. In the second situation, this means \emph{a
priori} that either $y=h_n^+$ or $y=h_n^-$. But we know that $\gamma$ is
part of a composition that is equal to either the source or the target of
$\alpha_k$ (see \ref{paragr:orientals}) and thus, $y$ is part of a composition that is equal to either the source or the target of $x=h^+_n$. Since no composition involving $h^-_n$ can be equal to $h^+_n$ (one could invoke the function introduced in \ref{prop:countingfunction}), this implies that $y=h_n^+$ and hence, $f(\gamma)$ is in the image of $j^+_n$. This goes for all generating cells of dimension $k-1$ of $\Or_k$ and we can recursively apply the same reasoning for generating cells of dimension $k-2$, then $k-3$ and so forth. Altogether, this proves that $\varphi$ factorizes through $j^+_n$. The case where $x=h^-_n$ and $\varphi$ factorizes through $j^-_n$ is symmetric.
\end{description}
\end{proof}
From these two lemmas, follows the important proposition below.
......@@ -275,7 +289,7 @@ higher than $1$.
(a',p) &\mapsto a'.
\end{aligned}
\]
This is special case of the more general notion of slice $\oo$\nbd{}categories introduced in \ref{paragr:slices}. In particular, given an $\oo$\nbd{}category $X$ and an $\oo$\nbd{}functor $f : X \to A$, we have defined the $\oo$\nbd{}category $X/a$ and the $\oo$\nbd{}functor
This is a special case of the more general notion of slice $\oo$\nbd{}categories introduced in \ref{paragr:slices}. In particular, given an $\oo$\nbd{}category $X$ and an $\oo$\nbd{}functor $f : X \to A$, we have defined the $\oo$\nbd{}category $X/a$ and the $\oo$\nbd{}functor
\[
f/a : X/a \to A/a
\]
......@@ -286,7 +300,19 @@ higher than $1$.
A/a \ar[r,"\pi_{a}"] &A.
\end{tikzcd}
\]
More explicitly, the $n$\nbd{}cells of $X/a$ can be described as pairs $(x,p)$ where $x$ is an $n$\nbd{}cell of $X$ and $p$ is an arrow of $A$ of the form $ p : f(\trgt_0(x)) \to a$. When $n>1$, the source and target of such an $n$\nbd{}cell are given by
More explicitly, the $n$\nbd{}cells of $X/a$ can be described as pairs $(x,p)$
where $x$ is an $n$\nbd{}cell of $X$ and $p$ is an arrow of $A$ of the form
\[
p : f(x)\to p \text{ if }n=0
\]
and
\[
p : f(\trgt_0(x)) \to a \text{ if }n>0.
\]
\emph{From now on, let us use the convention that $\trgt_0(x)=x$ when $x$ is a
$0$\nbd{}cell of $X$}.
When $n>0$, the source and target of an $n$\nbd{}cell $(x,p)$ of $X/a$ are given by
\[
\src((x,p))=(\src(x),p) \text{ and } \trgt((x,p))=(\trgt(x),p).
\]
......@@ -321,7 +347,7 @@ higher than $1$.
\[
\colim_{a \in A} (X/{a}) \to X.
\]
This map is natural in $X$ in the following sense. Let $f' : X' \to A$ be another $\oo$\nbd{}functor and let
Let $f' : X' \to A$ be another $\oo$\nbd{}functor and let
\[
\begin{tikzcd}
X \ar[rr,"g"] \ar[dr,"f"'] && X' \ar[dl,"f'"] \\
......@@ -383,7 +409,7 @@ higher than $1$.
\]
is commutative. This proves the existence part of the universal property.
Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangles commute for every object $a$ of $A$ and let $x$ be a $n$\nbd{}cell of $X$. Since the triangle
Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangles commute for every object $a$ of $A$ and let $x$ be an $n$\nbd{}cell of $X$. Since the triangle
\[
\begin{tikzcd}
X/f(\trgt_0(x)) \ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r] & X \ar[d,"\phi'"] \\
......@@ -503,8 +529,8 @@ Beware that in the previous corollary, we did \emph{not} suppose that $X$ was fr
and consider the following commutative square of $\ho(\oo\Cat^{\folk})$
\begin{equation}\label{comsquare}
\begin{tikzcd}
\displaystyle\hocolim_{a \in A}(P/a) \ar[d] \ar[r] & \displaystyle\colim_{a \in A}(P/a) \ar[d] \\
\displaystyle\hocolim_{a \in A}(X/a) \ar[r] & \displaystyle\colim_{a \in A}(X/a)
\displaystyle\hocolim^{\folk}_{a \in A}(P/a) \ar[d] \ar[r] & \displaystyle\colim_{a \in A}(P/a) \ar[d] \\
\displaystyle\hocolim^{\folk}_{a \in A}(X/a) \ar[r] & \displaystyle\colim_{a \in A}(X/a)
\end{tikzcd}
\end{equation}
where the vertical arrows are induced by the arrows
......@@ -563,7 +589,7 @@ We now recall an important theorem due to Thomason.
\]
is canonically isomorphic to the homotopy colimit functor
\[
\hocolim_A : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th}).
\hocolim^{\Th}_A : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th}).
\]
\end{theorem}
\begin{proof}
......@@ -572,7 +598,7 @@ We now recall an important theorem due to Thomason.
\begin{corollary}\label{cor:thomhmtpycol}
Let $A$ be a $1$\nbd{}category. The canonical map
\[
\hocolim_{a \in A}(A/a) \to A
\hocolim^{\Th}_{a \in A}(A/a) \to A
\]
induced by the co-cone $(A/a \to A)_{a \in \Ob(A)}$, is an isomorphism of $\ho(\Cat^{\Th})$.
\end{corollary}
......@@ -591,14 +617,14 @@ We now recall an important theorem due to Thomason.
\[
\int_{a \in A}A/a \to \int_{a \in A}k_{\sD_0}
\]
is a Thomason equivalence. An immediate computation shows that \[\int_{a \in A}k_{\sD_0} \simeq A.\] From the second part of Theorem \ref{thm:Thomason}, we have that
is a Thomason equivalence and an immediate computation shows that \[\int_{a \in A}k_{\sD_0} \simeq A.\] From the second part of Theorem \ref{thm:Thomason}, we have that
\[
\hocolim_{a \in A}(A/a) \simeq A.
\hocolim^{\Th}_{a \in A}(A/a) \simeq A.
\]
A thorough analysis of all the isomorphisms involved shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in \Ob(A)}$.
\end{proof}
\begin{remark}
It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}categories), we have \[\hocolim^{\Th}_{a \in A} (X/a) \simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.
It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}categories), we have \[\hocolim^{\Th}_{a \in A} (X/a) \simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation.
\end{remark}
Putting all the pieces together, we are now able to prove the awaited Theorem.
\begin{theorem}\label{thm:categoriesaregood}
......
......@@ -1356,7 +1356,11 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t
\begin{equation}
\Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)).
\end{equation}
Now let $\Delta_{\leq 2}$ be the full subcategory of $\Delta$ spanned by $[0]$, $[1]$ and $[2]$ and $i : \Delta_{\leq 2} \to \Delta$ the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$. Recall that the nerve of a (small) category is $2$-coskeletal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have
Now let $\Delta_{\leq 2}$ be the full subcategory of $\Delta$ spanned by
$[0]$, $[1]$ and $[2]$ and let $i : \Delta_{\leq 2} \to \Delta$ be the
canonical inclusion. This inclusion induces by pre-composition a functor $i^*
: \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* :
\Psh{\Delta_{\leq 2}} \to \Psh{\Delta}$. Recall that the nerve of a (small) category is $2$-coskeletal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have
\begin{align*}
\Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)) &\simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),i_* i^* (N_1(D)))\\
&\simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))).
......
......@@ -209,7 +209,12 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\begin{corollary}\label{cor:thomhmtpycocomplete}
For every $1 \leq n \leq \oo$, the localizer $(n\Cat^{\Th},\W_n^{\Th})$ is homotopy cocomplete (Definition \ref{def:cocompletelocalizer}).
\end{corollary}
Another consequence of Gagna's theorem is the following corollary.
We will speak of ``Thomason homotopy colimits'' and ``Thomason homotopy
cocartesian square'' for homotopy colimits and homotopy cocartesian squares in
the localizer $(n\Cat^{\Th},\W_n^{\Th})$.
Another consequence of Gagna's theorem is the following
corollary.
\begin{corollary}\label{cor:thomsaturated}
For every $1 \leq n \leq \oo$, the class $\W_n^{\Th}$ is saturated (\ref{paragr:loc}).
\end{corollary}
......@@ -557,8 +562,11 @@ For later reference, we put here the following trivial but important lemma, whos
\[
\gamma^{\folk} : \oo\Cat \to \Ho(\oo\Cat^{\folk})
\]
for the localization morphism.
It follows from the previous theorem and Theorem \ref{thm:cisinskiI} that the localizer $(\oo\Cat,\W_{\oo}^{\folk})$ is homotopy cocomplete.
for the localization morphism. It follows from the previous theorem and
Theorem \ref{thm:cisinskiI} that the localizer $(\oo\Cat,\W_{\oo}^{\folk})$
is homotopy cocomplete. We will speak of ``folk homotopy
colimits'' and ``folk homotopy cocartesian squares'' for homotopy colimits
and homotopy cocartesian squares in this localizer.
\end{paragr}
\begin{paragr}\label{paragr:folktrivialfib}
Using the set $\{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N} \}$ of generating folk cofibrations, we obtain that an $\oo$\nbd{}functor $F : C \to D$ is a \emph{folk trivial fibration} when:
......
......@@ -932,8 +932,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
In this section, we quickly review some aspects of the relation between
Quillen's theory of model categories and Grothendieck's theory of derivators.
We suppose that the reader is familiar with the former one and refer to the
standard textbooks on the subject (such as \cite{hovey2007model},
\cite{hirschhorn2009model} or \cite{dwyer1995homotopy}) for basic definitions
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
......@@ -952,7 +951,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{theorem}
\begin{theorem}[Cisinski]\label{thm:cisinskiII}
Let $\M$ and $\M'$ be two model categories and ${F : \M \to \M'}$ a left
Let $\M$ and $\M'$ be two model categories and let ${F : \M \to \M'}$ 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 $\LL F : \Ho(\M) \to \Ho(\M')$ is homotopy
......@@ -1034,7 +1033,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\emph{projective model structure on $\M(A)$}.
\end{paragr}
\begin{proposition}
Let $(\M,\W,\Cof,\Fib)$ be a cofibrantely generated model category. For
Let $(\M,\W,\Cof,\Fib)$ be a cofibrantly generated model category. For
every $u : A \to B$, the adjunction
\[
\begin{tikzcd}
......@@ -1099,7 +1098,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
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 pointwise equivalences and for
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.
......
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