Commit 01539760 authored by Leonard Guetta's avatar Leonard Guetta
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Encore des typos et du layout

parent 5df9566f
......@@ -25,7 +25,7 @@ In this section, we review some homotopical results on free
are the morphisms $f : G \to G'$ that are injective on objects and on arrows,
i.e. such that $f_0 : G_0 \to G_0'$ and $f_1 : G_1 \to G'_1$ are injective.
There is a ``underlying reflexive graph'' functor
There is an ``underlying reflexive graph'' functor
\[
U : \Cat \to \Rgrph,
\]
......@@ -239,7 +239,7 @@ From the previous proposition, we deduce the following very useful corollary.
L(C) \ar[r,"L(\gamma)"]& L(D)
\end{tikzcd}
\]
is a Thomason homotopy cocartesian.
is Thomason homotopy cocartesian.
\end{corollary}
\begin{proof}
Since the nerve $N$ induces an equivalence of op-prederivators
......@@ -272,11 +272,11 @@ From the previous proposition, we deduce the following very useful corollary.
By working a little more, we obtain the more general result stated
in the proposition below. Let us say that a morphism of reflexive
graphs $\alpha : A \to B$ is \emph{quasi-injective on arrows} when
for all $f$ and $g$ arrows of $A$, if
for all arrows $f$ and $g$ of $A$, if
\[
\alpha(f)=\alpha(g),
\]
then either $f=g$ or $f$ and $g$ are both units. In other words, $\alpha$
then either $f=g$, or $f$ and $g$ are both units. In other words, $\alpha$
never sends a non-unit arrow to a unit arrow and $\alpha$ never identifies two
non-unit arrows. It follows that if $\alpha$ is quasi-injective on arrows and
injective on objects, then it is also injective on arrows and hence, a
......@@ -304,7 +304,7 @@ From the previous proposition, we deduce the following very useful corollary.
L(C) \ar[r,"L(\gamma)"] &L(D)
\end{tikzcd}
\]
is Thomason homotopy cocartesian square.
is Thomason homotopy cocartesian.
\end{proposition}
\begin{proof}
The case where $\alpha$ or $\beta$ is both injective on objects and
......@@ -415,7 +415,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref{cor:hmtpysquaregraph}.
\end{example}
\begin{remark}
Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration% , since a Thomason homotopy
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
......@@ -456,7 +456,7 @@ 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 above square is Thomason homotopy cocartesian. Indeed, it
Then, this square is Thomason homotopy cocartesian. Indeed, it
obviously is the image of a cocartesian square in $\Rgrph$ by the
functor $L$ and since the source and target of $f$ are different,
the top map comes from a monomorphism of $\Rgrph$. Hence, we can
......@@ -744,7 +744,7 @@ In practice, we will use the following corollary.
\ref{prop:bisimplicialcocontinuous}.
\end{proof}
\section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}
We shall now describe a ``nerve'' for $2$-categories with values in bisimplicial
We shall now describe a ``nerve'' for $2$\nbd{}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}
......@@ -757,23 +757,23 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
\]
is the set of $k$-simplices of the nerve of $C$.
\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
the nerve of $2$\nbd{}categories. This makes sense since the nerve for categories
is the restriction of the nerve for $2$\nbd{}categories.
\item[-] For $2$\nbd{}categories, we refer to the $\comp_0$-composition of
$2$\nbd{}cells as the \emph{horizontal composition} and the $\comp_1$-composition
of $2$\nbd{}cells as the \emph{vertical composition}.
\item[-] For a $2$\nbd{}category $C$ and $x$ and $y$ objects of $C$, we denote by
\[
C(x,y)
\]
the category whose objects are the $1$-cells of $C$ with $x$ as source and
$y$ as target, and whose arrows are the $2$-cells of $C$ with $x$ as
$y$ as target, and whose arrows are the $2$\nbd{}cells of $C$ with $x$ as
$0$-source and $y$ as $0$-target. Composition is induced by vertical
composition in $C$.
\end{itemize}
\end{notation}
\begin{paragr}
Every $2$-category $C$ defines a simplicial object in $\Cat$,
Every $2$\nbd{}category $C$ defines a simplicial object in $\Cat$,
\[S(C): \Delta^{\op} \to \Cat,\] where, for each $n \geq 0$,
\[
S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1)
......@@ -795,7 +795,7 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
category.
\end{remark}
\begin{definition}
The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set
The \emph{bisimplicial nerve} of a $2$\nbd{}category $C$ is the bisimplicial set
$\binerve(C)$ defined as
\[
\binerve(C)_{n,m}:=N(S_n(C))_m,
......@@ -804,7 +804,7 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
\end{definition}
\begin{paragr}\label{paragr:formulabisimplicialnerve}
In other words, the bisimplicial nerve of $C$ is obtained by ``un-currying''
the functor $NS(C) : \Delta^{op} \to \Psh{\Delta}$.
the functor $NS(C) : \Delta^{\op} \to \Psh{\Delta}$.
Since the nerve $N$ commutes with products and sums, we obtain the formula
\begin{equation}\label{fomulabinerve}
......@@ -847,7 +847,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
\[
\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)
\]
of vertically composable $2$-cells of $C$, i.e.\ such that for
of vertically composable $2$\nbd{}cells of $C$, i.e.\ such that for
every $1 \leq i \leq k-1$, we have
\[
\src(\alpha_i)=\trgt(\alpha_{i+1}).
......@@ -879,7 +879,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
degeneracy operators are induced by the units for the vertical composition.
\end{paragr}
\begin{lemma}\label{lemma:binervehorizontal}
Let $C$ be a $2$-category. For every $n \geq 0$, we have
Let $C$ be a $2$\nbd{}category. For every $n \geq 0$, we have
\[
N(V_m(C))_n=(\binerve(C))_{n,m}.
\]
......@@ -897,7 +897,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
theory of bisimplicial sets.
\end{paragr}
\begin{lemma}\label{lemma:binervthom}
A $2$-functor $F : C \to D$ is a Thomason equivalence if and only if
A $2$\nbd{}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}
......@@ -910,9 +910,9 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
\cite[Théorème 3.13]{ara2020comparaison}.
\end{proof}
From this lemma, we deduce two useful criteria to detect Thomason equivalences
of $2$-categories.
of $2$\nbd{}categories.
\begin{corollary}\label{cor:criterionThomeqI}
Let $F : C \to D$ be a $2$-functor. If
Let $F : C \to D$ be a $2$\nbd{}functor. If
\begin{enumerate}[label=\alph*)]
\item $F_0 : C_0 \to D_0$ is a bijection,
\end{enumerate}
......@@ -924,10 +924,10 @@ of $2$-categories.
\]
induced by $F$ is a Thomason equivalence of $1$-categories,
\end{enumerate}
then $F$ is a Thomason equivalence of $2$-categories.
then $F$ is a Thomason equivalence of $2$\nbd{}categories.
\end{corollary}
\begin{proof}
By definition, for every $2$-category $C$ and every $m \geq 0$, we have
By definition, for every $2$\nbd{}category $C$ and every $m \geq 0$, we have
\[
(\binerve(C))_{\bullet,m} = NS(C).
\]
......@@ -936,9 +936,9 @@ of $2$-categories.
products.
\end{proof}
\begin{corollary}\label{cor:criterionThomeqII}
Let $F : C \to D$ be a $2$-functor. If for every $k \geq 0$,
Let $F : C \to D$ be a $2$\nbd{}functor. If for every $k \geq 0$,
\[V_k(F) : V_k(C) \to V_k(D)\] is a Thomason equivalence of $1$-categories,
then $F$ is a Thomason equivalence of $2$-categories.
then $F$ is a Thomason equivalence of $2$\nbd{}categories.
\end{corollary}
\begin{proof}
From Lemma \ref{lemma:binervehorizontal}, we now that for every $m \geq 0$,
......@@ -1071,8 +1071,8 @@ of $2$-categories.
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 whose target is a
Let $n,m \geq 0$. We denote by $A_{(m,n)}$ the free $2$\nbd{}category with only one
generating $2$\nbd{}cell whose source is a chain of length $m$ and whose target is a
chain of length $n$:
\[
\underbrace{\overbrace{\begin{tikzcd}[column sep=small, ampersand
......@@ -1092,7 +1092,7 @@ of $2$-categories.
for } 0\leq i \leq m-1 \\ g_{j+1} : B_j \to B_{j+1} &\text{ for } 0 \leq
j \leq n-1 %\\g_1 : A_0 \to B_1 & \\g_{n} : B_{n-1} \to A_m &
\end{cases}$
\item generating $2$-cell: $ \alpha : f_{m}\circ \cdots \circ f_1 \Rightarrow
\item generating $2$\nbd{}cell: $ \alpha : f_{m}\circ \cdots \circ f_1 \Rightarrow
g_n \circ \cdots \circ g_1$.
\end{itemize}
Notice that $A_{(1,1)}$ is nothing but $\sD_2$. We are going to prove that if
......@@ -1113,10 +1113,10 @@ of $2$-categories.
\ar[from=A,to=1-1,Rightarrow,"\alpha"]
\end{tikzcd}
\]
and has many non trivial $2$-cells, such as $f\comp_0 \alpha \comp_0 f$.
and has many non trivial $2$\nbd{}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 $2$-category $B^2\mathbb{N}$ and we have already seen that it
Note that when $m=0$ \emph{and} $n=0$, then the $2$\nbd{}category $A_{(0,0)}$ is
nothing but the $2$\nbd{}category $B^2\mathbb{N}$ and we have already seen that it
is \emph{not} \good{} (see \ref{paragr:bubble}).
\end{paragr}
\begin{paragr}
......@@ -1154,9 +1154,9 @@ of $2$-categories.
end,"\ulcorner"]
\end{tikzcd}
\]
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$\nbd{}functor that sends the unique
non-trivial $1$\nbd{}cell of $\Delta_1$ to the target of the generating
$2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong
$2$\nbd{}cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong
deformation retract and thus, a co-universal Thomason equivalence (Lemma
\ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $\Delta_n \to
A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is
......@@ -1176,7 +1176,7 @@ of $2$-categories.
end,"\ulcorner"]
\end{tikzcd}
\]
where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the
where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$\nbd{}functor that sends the
unique non trivial $1$\nbd{}cell of $\Delta_1$ to the source of the generating
$2$\nbd{}cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has
the homotopy type of a point.
......@@ -1190,9 +1190,9 @@ of $2$-categories.
end,"\ulcorner"]
\end{tikzcd}
\]
where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of
$\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This
$2$-functor is once again a folk cofibration, but it is \emph{not} in general
where $\tau$ is the $2$\nbd{}functor that sends the unique non-trivial $1$-cell of
$\Delta_1$ to the target of the generating $2$\nbd{}cell of $A_{(m,1)}$. This
$2$\nbd{}functor is once again a folk cofibration, but it is \emph{not} in general
a co-universal Thomason equivalence (it would be if we had made the hypothesis that
$m\neq 0$, but we did not). However, since we made the hypothesis that $n\neq
0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta_1 \to
......@@ -1207,7 +1207,7 @@ of $2$-categories.
Combined with the result of Paragraph \ref{paragr:bubble}, we have proved the
following proposition.
\begin{proposition}\label{prop:classificationAmn}
Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or
Let $m,n \geq 0$ and consider the $2$\nbd{}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
$K(\mathbb{Z},2)$.
......@@ -1546,7 +1546,7 @@ Let us now get into more sophisticated examples.
\begin{center}
\begin{tabular}{| l || c | c |}
\hline
$2$-category & \good{}? & homotopy type \\ \hline \hline
$2$\nbd{}category & \good{}? & homotopy type \\ \hline \hline
{
$\begin{tikzcd}
\bullet \ar[r,bend
......
......@@ -182,11 +182,11 @@ homotopy theory, polygraphs.
\pagestyle{fancy}
\fancyhf{}
\fancyfoot[C]{\thepage}
\fancyhead[RO]{INTRODUCTION}
\fancyhead[RO,LE]{INTRODUCTION}
\include{introduction}
\fancyhf{}
\fancyfoot[C]{\thepage}
\fancyhead[RO]{INTRODUCTION (FRANÇAIS)}
\fancyhead[RO,LE]{INTRODUCTION (FRANÇAIS)}
\include{introduction_fr}
\fancyhf{}
\fancyfoot[C]{\thepage}
......
......@@ -57,7 +57,7 @@ year={2020}
year={2020}
}
@article{ara2019quillen,
title={A {Q}uillen Theorem {B} for strict $\infty$\nbd{}categories},
title={A {Q}uillen {T}heorem {B} for strict $\infty$\nbd{}categories},
author={Ara, Dimitri},
journal={Journal of the London Mathematical Society},
volume={100},
......@@ -67,7 +67,7 @@ year={2020}
publisher={Wiley Online Library}
}
@article{batanin1998monoidal,
title={Monoidal globular categories as a natural environment for the theory of weak n\nbd{}categories},
title={Monoidal globular categories as a natural environment for the theory of weak $n$\nbd{}categories},
author={Batanin, Michael A.},
journal={Advances in Mathematics},
volume={136},
......@@ -137,7 +137,7 @@ publisher = "Elsevier"
publisher={Citeseer}
}
@article{cisinski2003images,
title={Images directes cohomologiques dans les cat{\'e}gories de modeles},
title={Images directes cohomologiques dans les cat{\'e}gories de mod{\`e}les},
author={Cisinski, Denis-Charles},
journal={Annales Math{\'e}matiques Blaise Pascal},
volume={10},
......@@ -170,7 +170,7 @@ publisher = "Elsevier"
year={1995}
}
@article{duskin2002simplicial,
title={Simplicial matrices and the nerves of weak n\nbd{}categories. {I}. Nerves of bicategories},
title={Simplicial matrices and the nerves of weak $n$\nbd{}categories. {I}. Nerves of bicategories},
author={Duskin, John W.},
journal={Theory and Applications of Categories},
volume={9},
......@@ -197,7 +197,7 @@ publisher = "Elsevier"
year={1995}
}
@article{eilenberg1954groups,
title={On the groups {H}($\pi$,n), {II}: Methods of computation},
title={On the groups ${H}(\pi,n)$, {II}: Methods of computation},
author={Eilenberg, Samuel and Mac Lane, Saunders},
journal={Annals of Mathematics},
pages={49--139},
......@@ -221,7 +221,7 @@ publisher = "Elsevier"
publisher={Springer}
}
@article{gagna2018strict,
title={Strict n-categories and augmented directed complexes model homotopy types},
title={Strict $n$\nbd{}categories and augmented directed complexes model homotopy types},
author={Gagna, Andrea},
journal={Advances in Mathematics},
volume={331},
......@@ -480,7 +480,7 @@ note={In preparation}
publisher={Springer}
}
@article{schreiber2013differential,
title={Differential cohomology in a cohesive infinity-topos},
title={Differential cohomology in a cohesive $\infty$\nbd{}topos},
author={Schreiber, Urs},
journal={arXiv preprint arXiv:1310.7930},
year={2013}
......
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