Commit ba4e0f82 authored by Leonard Guetta's avatar Leonard Guetta
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Modifs suggérés par Garner presque terminées

parent 24f2551e
......@@ -296,14 +296,14 @@ From the previous proposition, we deduce the following very useful corollary.
\item Either $\alpha$ or $\beta$ is injective on objects.
\item Either $\alpha$ or $\beta$ is quasi-injective on arrows.
\end{enumerate}
Then, the square
Then, the induced square of $\Cat$
\[
\begin{tikzcd}
L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\
L(C) \ar[r,"L(\gamma)"] &L(D)
\end{tikzcd}
\]
is a Thomason homotopy cocartesian square of $\Cat$.
is Thomason homotopy cocartesian square.
\end{proposition}
\begin{proof}
The case where $\alpha$ or $\beta$ is both injective on objects and
......@@ -408,16 +408,17 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
Then, this square is Thomason homotopy cocartesian in
$\Cat$. Indeed, it obviously is the image of a square of $\Rgrph$ by
Then, this square is Thomason homotopy cocartesian. Indeed, it obviously is the image of a square of $\Rgrph$ by
the functor $L$ and the morphism $i_1 : \sS_0 \to \sD_1$ comes from
a monomorphism of $\Rgrph$. Hence, we can apply Corollary
\ref{cor:hmtpysquaregraph}.
\end{example}
\begin{remark}
Since every free category is obtained by recursively adding generators
Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration% , since a Thomason homotopy
% cocartesian square in $\Cat$ is also so in $\oo\Cat$
and since every free category is obtained by recursively adding generators
starting from a set of objects (seen as a $0$-category), the previous example
yields another proof that \emph{free} (1-)categories are \good{} (which we
yields another proof that \emph{free} (1\nbd{})categories are \good{} (which we
already knew since we have seen that \emph{all} (1-)categories are \good{}).
\end{remark}
\begin{example}[Identifying two generators]
......@@ -434,7 +435,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\]
where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating
arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square
is Thomason homotopy cocartesian in $\Cat$.
is Thomason homotopy cocartesian.
Indeed, it is the image by the functor $L$ of a cocartesian square in
$\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the
morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply
......@@ -777,9 +778,8 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1)
\times \cdots \times C(x_{n-1},x_n).
\]
Note that for $n=0$, the above formula reads $S_0(C)=C_0$. The face
operators $\partial_i : S_{n}(C) \to S_{n-1}(C)$ are induced by horizontal
composition and the degeneracy operators $s_i : S_{n}(C) \to S_{n+1}(C)$ are
Note that for $n=0$, the above formula reads $S_0(C)=C_0$. For $n>0$, the face operators $\partial_i : S_{n}(C) \to S_{n-1}(C)$ are induced by horizontal
composition for $0 < i <n$ and by projection for $i=0$ or $n$. The degeneracy operators $s_i \colon S_{n}(C) \to S_{n+1}(C)$ are
induced by the units for the horizontal composition.
Post-composing $S(C)$ with the nerve functor $N : \Cat \to \Psh{\Delta}$, we
......@@ -952,47 +952,77 @@ of $2$-categories.
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).
\]
As we shall soon see, this morphism is an \emph{equivalence} of
op-prederivators. First, consider the triangle of functors
\end{paragr}
\begin{proposition}\label{prop:streetvsbisimplicial}
The morphism of op\nbd{}prederivators
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
is an equivalence of op\nbd{}prederivators.
\end{proposition}
\begin{proof}
Consider the following triangle of functors
\[
\begin{tikzcd}
2\Cat \ar[rr,"\binerve"] \ar[dr,"N"'] & & \Psh{\Delta\times\Delta} \ar[ld,"\delta^*"] \\
&\Psh{\Delta}.
\end{tikzcd}
\]
This triangle is \emph{not} commutative but it becomes commutative (up to an
isomorphism) after localization.
\end{paragr}
\begin{proposition}\label{prop:streetvsbisimplicial}
The triangle of morphisms of op-prederivators
\[
Even though it is \emph{not} commutative (even up to an isomorphism), it
follows from the results contained in \cite[Section2]{bullejos2003geometry}
that the induced triangle
\[
\begin{tikzcd}
\Ho(2\Cat^{\Th}) \ar[rr,"\overline{\binerve}"] \ar[dr,"\overline{N}"'] & & \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[ld,"\overline{\delta^*}"] \\
&\Ho(\Psh{\Delta})
\end{tikzcd}
\]
is commutative up to a canonical isomorphism.
\end{proposition}
\begin{proof}
It is a consequence of the results contained in \cite[Section
2]{bullejos2003geometry}.
\end{proof}
\begin{paragr}
Since $\overline{\delta^*}$ and $\overline{N}$ are equivalences of
is commutative up to a canonical isomorphism. The result follows then from the
fact that $\overline{\delta^*}$ and $\overline{N}$ are equivalences of
op-prederivators (Proposition \ref{prop:diageqderivator} and Theorem
\ref{thm:gagna} respectively), it follows from the previous proposition that
the morphism
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
is an \emph{equivalence} of op-prederivators.
\ref{thm:gagna} respectively).
\end{proof}
% As we shall soon see, this morphism is an \emph{equivalence} of
% op-prederivators. First, consider the triangle of functors
% \[
% \begin{tikzcd}
% 2\Cat \ar[rr,"\binerve"] \ar[dr,"N"'] & & \Psh{\Delta\times\Delta} \ar[ld,"\delta^*"] \\
% &\Psh{\Delta}.
% \end{tikzcd}
% \]
% This triangle is \emph{not} commutative but it becomes commutative (up to an
% isomorphism) after localization.
% \end{paragr}
%
% \[
% \begin{tikzcd}
% \Ho(2\Cat^{\Th}) \ar[rr,"\overline{\binerve}"] \ar[dr,"\overline{N}"'] & & \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[ld,"\overline{\delta^*}"] \\
% &\Ho(\Psh{\Delta})
% \end{tikzcd}
% \]
% is commutative up to a canonical isomorphism.
% \end{proposition}
% \begin{proof}
% It is a consequence of the results contained in \cite[Section
% 2]{bullejos2003geometry}.
% \end{proof}
% \begin{paragr}
% Since $\overline{\delta^*}$ and $\overline{N}$ are equivalences of
% op-prederivators (Proposition \ref{prop:diageqderivator} and Theorem
% \ref{thm:gagna} respectively), it follows from the previous proposition that
% the morphism
% \[
% \overline{\binerve} : \Ho(2\Cat^{\Th}) \to
% \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
% \]
% is an \emph{equivalence} of op-prederivators.
\begin{paragr}
\todo{enlever l'environnement paragr}
From Proposition \ref{prop:streetvsbisimplicial}, we deduce the proposition
below which contains two useful criteria to detect Thomason homotopy
cocartesian squares of $2\Cat$.
\end{paragr}
\end{paragr}
\begin{proposition}\label{prop:critverthorThomhmtpysquare}
Let
\begin{equation}\tag{$\ast$}\label{coucou}\begin{tikzcd}
......@@ -1001,24 +1031,24 @@ of $2$-categories.
\end{tikzcd}\end{equation}
be a square in $2\Cat$ satisfying at least one of the two following conditions:
\begin{enumerate}[label=(\alph*)]
\item For every $n\geq 0$, the square
\item For every $n\geq 0$, the commutative square of $\Cat$
\[
\begin{tikzcd}
V_{n}(A) \ar[r,"V_{n}(u)"]\ar[d,"V_{n}(f)"'] & V_n(B) \ar[d,"V_n(g)"] \\
V_n(C) \ar[r,"V_n(v)"] & V_n(D)
\end{tikzcd}
\]
is a Thomason homotopy cocartesian square of $\Cat$,
is Thomason homotopy cocartesian,
\item For every $n\geq 0$, the square
\item For every $n\geq 0$, the commutative square of $\Cat$
\[
\begin{tikzcd}
S_{n}(A) \ar[r,"S_{n}(u)"]\ar[d,"S_{n}(f)"'] & S_n(B) \ar[d,"S_n(g)"] \\
S_n(C) \ar[r,"S_n(v)"] & S_n(D)
\end{tikzcd}
\] is a Thomason homotopy cocartesian square of $\Cat$.
\] is Thomason homotopy cocartesian.
\end{enumerate}
Then, square \eqref{coucou} is a Thomason homotopy cocartesian in $2\Cat$.
Then, square \eqref{coucou} is Thomason homotopy cocartesian.
\end{proposition}
\begin{proof}
This is an immediate consequence of Proposition
......
......@@ -286,9 +286,11 @@ higher than $1$.
Let $A$ be a $1$\nbd{}category and $a$ an object of $A$. Recall that we write $A/a$ for the slice $1$\nbd{}category of $A$ over $a$, that is the $1$\nbd{}category whose description is as follows:
\begin{itemize}[label=-]
\item an object of $A/a$ is a pair $(a', p : a' \to a)$ where $a'$ is an object of $A$ and $p$ is an arrow of $A$,
\item an arrow $(a',p) \to (a'',p')$ of $A/a$ is an arrow $ q : a' \to a''$ of $A$ such that $p'\circ q = p$,
\item an arrow of $A/a$ is a pair $(q,p : a' \to a)$ where $p$ is an arrow of
$A$ and $q$ is an arrow of $A$ of the form $q : a'' \to a'$. The target of
$(q,p)$ is given by $(a',p)$ and the source by $(a'',p\circ q)$. % $ q : a' \to a''$ of $A$ such that $p'\circ q = p$.
\end{itemize}
and we write $\pi_a$ for the canonical forgetful functor
We write $\pi_a$ for the canonical forgetful functor
\[
\begin{aligned}
\pi_{a} : A/a &\to A \\
......@@ -433,7 +435,9 @@ higher than $1$.
we obtain that every $1$\nbd{}category $A$ is (canonically isomorphic to) the colimit
\[
\colim_{a \in A} (A/a).
\]
\]
% In other words, this simply say that the colimit of the Yoneda embedding $A \to
% \Psh{A}$ is the terminal presheaves
We now proceed to prove that this colimit is homotopic with respect to
the folk weak equivalences.
\end{paragr}
......@@ -532,24 +536,32 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
Beware that in the previous corollary, we did \emph{not} suppose that $X$ was free.
\begin{proof}
Let $P$ be a free $\omega$-category and $g : P \to X$ a folk trivial fibration
and consider the following commutative square of $\ho(\oo\Cat^{\folk})$
and consider the following commutative diagram of $\ho(\oo\Cat^{\folk})$
\begin{equation}\label{comsquare}
\begin{tikzcd}
\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)
\displaystyle\hocolim^{\folk}_{a \in A}(P/a) \ar[d] \ar[r] & \displaystyle\colim_{a \in A}(P/a) \ar[d] \ar[r] & P \ar[d]\\
\displaystyle\hocolim^{\folk}_{a \in A}(X/a) \ar[r] & \displaystyle\colim_{a \in A}(X/a) \ar[r] & X
\end{tikzcd}
\end{equation}
where the vertical arrows are induced by the arrows
where the middle and most left vertical arrows are induced by the arrows
\[
g/a : P/a \to X/a.
g/a : P/a \to X/a,
\]
Since trivial fibrations are stable by pullback, $g/a$ is a trivial fibration. This proves that the left vertical arrow of square (\ref{comsquare}) is an isomorphism.
and the most right vertical arrow is induced by $g$.
Since trivial fibrations are stable by pullback, $g/a$ is a trivial fibration.
This proves that the most left vertical arrow of diagram \eqref{comsquare} is an isomorphism.
Moreover, from Paragraph \ref{paragr:unfolding} and Lemma \ref{lemma:colimslice}, we deduce that the right vertical arrow of (\ref{comsquare}) can be identified with the image of $g : P \to X$ in $\ho(\omega\Cat)$ and hence is an isomorphism.
Finally, from Proposition \ref{prop:sliceiscofibrant} and Corollary \ref{cor:cofprojms}, we deduce that the top horizontal arrow of (\ref{comsquare}) is an isomorphism.
From Proposition \ref{prop:sliceiscofibrant} and Corollary
\ref{cor:cofprojms}, we deduce that the arrow \[\hocolim_{a \in
A}^{\folk}(P/a)\to \colim_{a \in A}(P/a)\] is an isomorphism. Moreover, from Lemma \ref{lemma:colimslice}, we know that the arrows
\[\colim_{a \in A}(P/a)\to P\] and \[\colim_{a \in A}(X/a)\to X\] are
isomorphisms.
By an immediate 2-out-of-3 property, this proves the result.
Finally, since $g$ is a folk weak equivalence, the most right vertical arrow of diagram \eqref{comsquare} is an
isomorphism and by an immediate 2-out-of-3 property this proves
that all arrows of \eqref{comsquare} are isomorphisms. In particular, so is
the composition of the two bottom horizontal arrows, which is what we desired
to show.
\end{proof}
We now move on to the next step needed to prove that every $1$\nbd{}category is \good{}. For that purpose, let us recall a construction commonly referred to as the ``Grothendieck construction''.
\begin{paragr}
......
......@@ -218,7 +218,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{corollary}
We will speak of ``Thomason homotopy colimits'' and ``Thomason homotopy
cocartesian squares'' for homotopy colimits and homotopy cocartesian squares in
the localizer $(n\Cat^{\Th},\W_n^{\Th})$.
the localizer $(n\Cat^{\Th},\W_n^{\Th})$. (See also \ref{paragr:thomhmtpycol} below.)
Another consequence of Gagna's theorem is the following
corollary.
......@@ -232,7 +232,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$, by Ara and Maltsiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.
\end{remark}
By definition, for all $1 \leq n \leq m \leq \omega$, the canonical inclusion \[n\Cat \hookrightarrow m\Cat\] sends the Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivators $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})$.
\begin{proposition}
\begin{proposition}\label{prop:nthomeqder}
For all $1 \leq n \leq m \leq \omega$, the canonical morphism
\[
\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})
......@@ -248,6 +248,21 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{tikzcd}
\]
\end{proof}
\begin{paragr}\label{paragr:thomhmtpycol}
It follows from the previous proposition that for all $1 \leq n \leq m \leq
\omega$, the morphism $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})$ of
op\nbd{}prederivators is homotopy cocontinuous and reflects homotopy
colimits (in an obvious sense). Hence, given a diagram $d : I
\to n\Cat$ with $n>0$, we can harmlessly use the notation
\[
\hocolim^{\Th}_{i \in I}(d)
\]
for both the Thomason homotopy colimits in $n\Cat$ and in $\oo\Cat$ (or any
$m\Cat$ with $n\leq m$). Similarly, a commutative square of
$n\Cat$ is Thomason homotopy cocartesian in $n\Cat$ if and
only if it is so in $\oo\Cat$. Hence, there is really no ambiguity when simply
calling such a square \emph{Thomason homotopy cocartesian}.
\end{paragr}
\section{Tensor product and oplax transformations}
Recall that $\oo\Cat$ can be equipped with a monoidal product $\otimes$, introduced by Al-Agl and Steiner in \cite{al1993nerves} and by Crans in \cite{crans1995combinatorial}, commonly referred to as the \emph{Gray tensor product}. The implicit reference for this section is \cite[Appendices A and B]{ara2016joint}.
\begin{paragr}
......@@ -291,7 +306,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\begin{tikzcd}
& Y \\
X \ar[ru,"u"] \ar[r,"\alpha"] \ar[rd,"v"']& \homlax(\sD_1,Y) \ar[u,"\pi_0^Y"'] \ar[d,"\pi_1^Y"] \\
& Y
& Y,
\end{tikzcd}
\]
where $\pi^Y_0$ and $\pi^Y_1$ are induced by the two $\oo$\nbd{}functors $\sD_0 \to \sD_1$ and where we implicitly used the isomorphism $\homlax(\sD_0,Y)\simeq Y$, is commutative.
......
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