Commit 45858165 authored by Leonard Guetta's avatar Leonard Guetta
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it's getting closer

parent 977ae1e8
......@@ -436,7 +436,7 @@ Diagonal weak equivalences are not the only interesting weak equivalences for bi
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).
\]
\end{paragr}
\begin{proposition}
\begin{proposition}\label{prop:bisimplicialcocontinuous}
The morphisms of op-prederivators
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
......@@ -475,30 +475,54 @@ Diagonal weak equivalences are not the only interesting weak equivalences for bi
\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta})
\]
are homotopy cocontinuous.
On the other hand, the obvious identity $\delta^*=\delta^* \circ
\mathrm{id}_{\Psh{\Delta\times\Delta}}$ implies that we have the commutative
triangles
\[
\begin{tikzcd}
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \ar[r] \ar[rd,"\overline{\delta^*}"']&
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\ar[d,"\overline{\delta^*}"] \\
&\Ho(\Psh{\Delta})
\end{tikzcd}
\]
and
\[
\begin{tikzcd}
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \ar[r] \ar[rd,"\overline{\delta^*}"']&
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\ar[d,"\overline{\delta^*}"] \\
&\Ho(\Psh{\Delta}).
\end{tikzcd}
\]
The result follows then from the fact that $\overline{\delta^*} :
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})$ is an
equivalence of op-prederivators.
\end{proof}
In practise, we will use the following corollary.
\begin{corollary}\label{cor:bisimplicialsquare}
Let
\begin{equation}\tag{$\ast$}\label{kiki}
\[
\begin{tikzcd}
A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}
\end{equation}
\]
be a square in the category of bisimplicial sets satisfying either of the following conditions:
\begin{enumerate}[label=(\alph*)]
\item For every $n\geq 0$, the square of simplicial sets
\[
\[
\begin{tikzcd}
A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\
C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}
A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
\end{tikzcd}
\]
\]
is homotopy cocartesian.
\item For every $n\geq 0$, the square of simplicial sets
\[
\[
\begin{tikzcd}
A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\
C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}
\end{tikzcd}
\]
is homotopy cocartesian.
......@@ -513,19 +537,39 @@ Diagonal weak equivalences are not the only interesting weak equivalences for bi
is a homotopy cocartesian square of simplicial sets.
\end{corollary}
\begin{proof}
\todo{Écrire une preuve ?}
\end{proof}
From \cite[Corollary 10.3.10(i)]{groth2013book} we know that the square of
bisimplicial sets
\[
\begin{tikzcd}
A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}
\]
is homotopy cocartesian with respect to the vertical weak
equivalences if and only if for every $n\geq 0$, the square
\[
\begin{tikzcd}
A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
\end{tikzcd}
\]
is a homotopy cocartesian square of simplicial sets and similarly for
horizontal weak equivalences. The result follows then
from Proposition \ref{prop:bisimplicialcocontinuous}.
\end{proof}
\section{Bisimplicial nerve for 2-categories}
We shall now describe a ``nerve'' for $2$-categories with values in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the Street nerve from section \todo{ref}.
We shall now describe a ``nerve'' for $2$-categories with values in bisimplicial
sets and recall a few results that shows that this nerve is, in some sense,
equivalent to the nerve defined in \ref{paragr:nerve}.
\begin{notation}
\begin{itemize}
\item[-] Once again, we will use the notation $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for the usual nerve of categories. Moreover, using the usual convention for the set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then
\item[-] Once again, we write $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for the usual nerve of categories. Moreover, using the usual convention for the set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then
\[
N(C)_k
\]
is the set of $k$-simplices of the nerve of $C$.
\item[-] Similarly, we will use the notation $N : 2\Cat \to \Psh{\Delta}$ instead of $N_2$ for the nerve of $2$-categories \todo{ref interne}. This makes sense since the nerve for categories is the restriction of the nerve for $2$-categories.
\item[-] For $2$-categories, we refer to the $\comp_0$-composition of $2$-cells as the \emph{horizontal composition} and the $\comp_1$-composition of $2$-cells as the \emph{vertical composition}. \todo{Uniformiser}
\item[-] Similarly, we write $N : 2\Cat \to \Psh{\Delta}$ instead of $N_2$ for the nerve of $2$-categories. This makes sense since the nerve for categories is the restriction of the nerve for $2$-categories.
\item[-] For $2$-categories, we refer to the $\comp_0$-composition of $2$-cells as the \emph{horizontal composition} and the $\comp_1$-composition of $2$-cells as the \emph{vertical composition}.
\item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by
\[
C(x,y)
......@@ -541,7 +585,7 @@ We shall now describe a ``nerve'' for $2$-categories with values in bisimplicial
H(C)_n:= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1) \times \cdots \times C(x_{n-1},x_n).
\]
Note that for $n=0$, the above formula reads $H(C)_0=\Ob(C)$.
The face operators$ \partial_i : H(C)_{n} \to H(C)_{n-1}$ are induced by horizontal composition and the degeneracy operators $s_i : H(C)_{n} \to H(C)_{n+1}$ are induced by the units for the horizontal composition.
The face operators $\partial_i : H(C)_{n} \to H(C)_{n-1}$ are induced by horizontal composition and the degeneracy operators $s_i : H(C)_{n} \to H(C)_{n+1}$ are induced by the units for the horizontal composition.
Post-composing $H(C)$ with the nerve functor $N : \Cat \to \Psh{\Delta}$, we obtain a functor
\[
......@@ -576,12 +620,11 @@ More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting sche
\ar[from=K,to=L,Rightarrow]
\end{tikzcd}\right.}_{ n }.
\]
\todo{parenthèses moches dans le diagramme. Mettre dessins en petites dimensions des opérateurs de faces ?}
\end{paragr}
In the definition of the bisimplicial nerve of a $2$-category we gave, we have priviledged one direction of the bisimplicial set over the other. We now give another definition of the bisimplicial nerve using the other direction.
In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, we have priviledged one direction of the bisimplicial set over the other. We now give another definition of the bisimplicial nerve using the other direction.
\begin{paragr}
Let $C$ be a $2$-category. For every $k \geq 1$, we define a $1$-category $V(C)_k$ in the following fashion:
\begin{itemize}
Let $C$ be a $2$\nbd{}category. For every $k \geq 1$, we define a $1$\nbd{}category $V(C)_k$ in the following fashion:
\begin{itemize}[label=-]
\item The objects of $V(C)_k$ are the objects of $C$.
\item A morphism $\alpha$ is a sequence
\[
......@@ -595,7 +638,7 @@ In the definition of the bisimplicial nerve of a $2$-category we gave, we have p
\[
\src(\alpha):=\src_0(\alpha_1)\text{ and }\trgt(\alpha):=\trgt_0(\alpha_1).
\]
(Note that we could have used any of the $\alpha_i$ instead of $\alpha_1$ since they all have the same $0$-source and $0$-target.)
(Note that we could have used any of the $\alpha_i$ instead of $\alpha_1$ since they all have the same $0$\nbd{}source and $0$\nbd{}target.)
\item Composition is given by
\[
(\alpha_1,\alpha_2,\cdots,\alpha_k)\circ(\beta_1,\beta_2,\cdots,\beta_k):=(\alpha_1\comp_0\beta_1,\alpha_2\comp_0\beta_2,\cdots,\alpha_k\comp_0\beta_k)
......@@ -605,7 +648,9 @@ In the definition of the bisimplicial nerve of a $2$-category we gave, we have p
(1^2_x,\cdots, 1^2_x).
\]
\end{itemize}
For $k=0$, we define $V(C)_0$ to be the category obtained from $C$ by simply forgetting the $2$-cells (\todo{Faire le lien avec le tronqué bête.}). The correspondance $n \mapsto V(C)_n$ defines to a simplicial object in $\Cat$
For $k=0$, we define $V(C)_0$ to be the category obtained from $C$ by simply
forgetting the $2$\nbd{}cells (which is nothing but $\tau^{s}_{\leq 1}(C)$
with the notations of \ref{paragr:ncat}). The correspondance $n \mapsto V(C)_n$ defines to a simplicial object in $\Cat$
\[
V(C) : \Delta^{\op} \to \Cat,
\]
......@@ -631,11 +676,13 @@ In the definition of the bisimplicial nerve of a $2$-category we gave, we have p
A $2$-functor $F : C \to D$ is a Thomason equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets.
\end{lemma}
\begin{proof}
It follows from what is shown in \cite[Section 2.1 and Theorem 2.7]{bullejos2003geometry} that there is weak equivalence of simplicial sets
It follows from what is shown in \cite[Section 2]{bullejos2003geometry} that
there is a zig-zag of weak equivalence of simplicial sets
\[
\delta^*(\binerve(C)) \to N(C)
\delta^*(\binerve(C)) \leftarrow \cdots \rightarrow N(C)
\]
which is natural in $C$. This implies what we wanted to show.
which is natural in $C$. This implies what we wanted to show. See also
\cite[Théorème 3.13]{ara2020comparaison}.
\end{proof}
From this lemma, we deduce two useful criteria to detect Thomason equivalences of $2$-categories.
\begin{corollary}\label{cor:criterionThomeqI}
......@@ -672,24 +719,27 @@ Let $F : C \to D$ be a $2$-functor. If for every $k \geq 0$,
The result follows them from Lemma \ref{bisimpliciallemma}.
\end{proof}
\begin{paragr}
It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve induces a morphism of op-prederivators
It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve induces a morphism of op\nbd{}prederivators
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}).
\]
This morphism is in fact an \emph{equivalence} of op-prederivators as we shall soon see. In order to do that, consider the triangle of functors
As we shall soon see, this morphism is an \emph{equivalence} of
op-prederivators. First, consider the triangle of functors
\[
\begin{tikzcd}
2\Cat \ar[rr,"\binerve"] \ar[dr,"N"] & & \Psh{\Delta\times\Delta} \ar[ld,"\delta^*"] \\
&\Psh{\Delta}
\end{tikzcd}.
2\Cat \ar[rr,"\binerve"] \ar[dr,"N"'] & & \Psh{\Delta\times\Delta} \ar[ld,"\delta^*"] \\
&\Psh{\Delta}.
\end{tikzcd}
\]
This triangle is \emph{not} commutative. However, the next proposition tells us that it becomes commutative (up to an isomorphism) after localization.
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
\[
\begin{tikzcd}
\Ho(2\Cat^{\Th}) \ar[rr,"\overline{\binerve}"] \ar[dr,"\overline{N}"] & & \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[ld,"\overline{\delta^*}"] \\
\Ho(2\Cat^{\Th}) \ar[rr,"\overline{\binerve}"] \ar[dr,"\overline{N}"'] & & \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[ld,"\overline{\delta^*}"] \\
&\Ho(\Psh{\Delta})
\end{tikzcd}
\]
......@@ -705,7 +755,7 @@ It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve in
\]
is an \emph{equivalence} of op-prederivators.
From Proposition \ref{prop:streetvsbisimplicial}, we also deduce the proposition below which contains two useful critera to detect Thomason homotopy cocartesian square of $2\Cat$.
From Proposition \ref{prop:streetvsbisimplicial}, we deduce the proposition below which contains two useful critera to detect Thomason homotopy cocartesian square of $2\Cat$.
\end{paragr}
\begin{proposition}
Let
......@@ -739,7 +789,20 @@ This is an immediate consequence of Proposition \ref{prop:streetvsbisimplicial}
\end{proof}
\section{Zoology of $2$-categories : Basic examples}
Before embarking on computations of homology and homotopy types of $2$-categories, let us recall the \todo{rappeler le corollaire 4.5.6 ?}
\begin{paragr}
Before embarking on computations of homology and homotopy types of
$2$\nbd{}categories, let us recall the following particular case of Corollary
\ref{cor:usefulcriterion}. Suppose given a cocartesian square
\[
\begin{tikzcd}
A \ar[r,"u"] \ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}
\]
of 2\nbd{}categories. If $A$,$B$ and $C$ are free and \good{}, if at least
$u$ or $f$ is a folk cofibration and if the square is Thomason homotopy
cocartesian, then $D$ is \good{}.
\end{paragr}
\begin{paragr}
Let $n,m \geq 0$. We denote by $A_{(m,n)}$ the free $2$-category with only one generating $2$-cell whose source is a chain of length $m$ and its target a chain of length $n$:
\[
......@@ -747,7 +810,10 @@ Before embarking on computations of homology and homotopy types of $2$-categorie
More formally, $A_{(m,n)}$ is described in the following way:
\begin{itemize}[label=-]
\item generating $0$-cells: $A_0,\cdots, A_m$, $B_1,\cdots,B_{n-1}$
\item generating $1$-cells: $\begin{cases} f_{i+1} : A_i \to A_{i+1} & \text{ for } 0\leq i \leq m-1 \\ g_{j+1} : B_j \to B_{j+1} &\text{ for } 1 \leq j \leq n-2 \\g_1 : A_0 \to B_1 & \\g_{n} : B_{n-1} \to A_m &
\end{itemize}
(And for convenience, we also set $B_0:=A_0$ and $B_n:=A_m$.)
\begin{itemize}
\item generating $1$-cells: $\begin{cases} f_{i+1} : A_i \to A_{i+1} & \text{ for } 0\leq i \leq m-1 \\ g_{j+1} : B_j \to B_{j+1} &\text{ for } 0 \leq j \leq n-1 %\\g_1 : A_0 \to B_1 & \\g_{n} : B_{n-1} \to A_m &
\end{cases}$
\item generating $2$-cell: $ \alpha : f_{m}\circ \cdots \circ f_1 \Rightarrow g_n \circ \cdots \circ g_1$.
\end{itemize}
......@@ -759,10 +825,11 @@ Before embarking on computations of homology and homotopy types of $2$-categorie
\]
and has many non trivial $2$-cells, such as $f\comp_0 \alpha \comp_0 f$.
Note that when $m=0$ \emph{and} $n=0$, then the $2$-category $A_{(0,0)}$ is nothing but the commutative monoid $\mathbb{N}$ seen as a $2$-category (\ref{paragr:bubble}) and we have already seen that it is \emph{not} \good{}. We shall refer to this $2$-category as the \emph{bubble}.
Note that when $m=0$ \emph{and} $n=0$, then the $2$-category $A_{(0,0)}$ is
nothing but the $2$-category $B^2\mathbb{N}$ (\ref{paragr:bubble}) and we have already seen that it is \emph{not} \good{}.
\end{paragr}
\begin{paragr}
Recall that for $n\geq 0$, we denote by $\Delta_n$ the linear order ${0 \leq \cdots \leq n}$ seen as a small category. Let $i : \Delta_1 \to \Delta_n$ be the unique functor such that
For $n\geq 0$, we write $\Delta_n$ for the linear order ${0 \leq \cdots \leq n}$ seen as a small category. Let $i : \Delta_1 \to \Delta_n$ be the unique functor such that
\[
i(0)=0 \text{ and } i(1)=n.
\]
......@@ -818,11 +885,50 @@ For any $n \geq 0$, consider the following cocartesian square
\end{paragr}
Combined with the result of Paragraph \ref{paragr:bubble}, we have proven the following proposition.
\begin{proposition}
Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or $n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$.
Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or $n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$\nbd{}space.
\end{proposition}
\section{Zoology of $2$-categories: A stub of a criterion}
\section{``Bubble-free'' $2$-categories}
\section{Zoology of $2$-categories: more examples}
\section{The ``Bubble-free'' conjecture}
\begin{definition}
Let $C$ be a $2$\nbd{}category. A \emph{bubble} (in $C$) is a $2$\nbd{}cell
$x$ of
$C$ such that:
\begin{itemize}[label=-]
\item $x$ is not a unit,
\item $\src_0(x)=\trgt_0(x)$,
\item $\trgt(x)=\src(x)=1_{\src_0(x)}$.
\end{itemize}
\end{definition}
\begin{paragr}
In picture, a bubble $x$ is a $2$\nbd{}cell of the form
\[
\begin{tikzcd}
A \ar[r,bend left=75,"1_A",""{name=A,below}] \ar[r,bend
right=75,"1_A"',pos=21/40,""{name=B,above}] &A,
\ar[from=A,to=B,"x",Rightarrow]
\end{tikzcd}
\]
where $A=\src_0(x)=\trgt_0(x)$.
\end{paragr}
\begin{definition}\label{def:bubblefree2cat}
A $2$\nbd{}category is said to be \emph{bubble-free} if it has no bubbles.
\end{definition}
\begin{paragr}
The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is
$B^2\mathbb{N}$. In fact, this $2$\nbd{}category classifies bubbles in the
sense that the functor
\begin{align*}
2\Cat &\to \Set \\
C &\mapsto \Hom_{2\Cat}(B^2\mathbb{N},C)
\end{align*}
is canonically isomorphic to the functor that sends a $2$\nbd{}category to its set of bubbles.
\todo{À finir}
\end{paragr}
\begin{conjecture}
A free $2$\nbd{}category is \good{} if and only if it is bubble-free.
\end{conjecture}
%%% Local Variables:
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No preview for this file type
......@@ -33,6 +33,12 @@
year={2018},
publisher={Elsevier}
}
@article{ara2020comparaison,
title={Comparaison des nerfs $ n $-cat{\'e}goriques},
author={Ara, Dimitri and Maltsiniotis, Georges},
journal={arXiv preprint arXiv:2010.00266},
year={2020}
}
@article{ara2020theoreme,
title={Un th{\'e}or{\`e}me {A} pour les $\infty$-cat{\'e}gories strictes {II} : la preuve $\infty$-cat{\'e}gorique},
author={Ara, Dimitri and Maltsiniotis, Georges},
......
......@@ -35,7 +35,7 @@
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
......
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