Commit a2823d7f authored by Leonard Guetta's avatar Leonard Guetta
Browse files

almost over

parent b1657859
......@@ -2,6 +2,7 @@
......@@ -356,10 +357,9 @@
\begin{exampleblock}{Important fact}
If $C$ is a free $\oo$\nbd{}category, the set of free generators of $C$
is uniquely determined from $C$.
If $C$ is a free $\oo$\nbd{}category, then there is a \emph{unique} set
of generating cells possible.
TODO : Laisser le bloc ci-dessus ?
\frametitle{Abelianization of $\oo$\nbd{}categories}
......@@ -703,14 +703,15 @@
\item<4-> Thus, for every $a \in A$, the $\oo$\nbd{}category $P/a$ is free.
\item<5-> Besides, the set of generating cells of $P/a$ ``vary naturally in a''.
\item<5-> Besides, the set of generating cells of $P/a$ is ``natural in a''.
\item<6-> Worked out properly (cf. thesis), this means that
\[P/{-} : A \to \oo\Cat\] is
cofibrant. \hfill CQFD
\begin{frame}\frametitle{In practice}
\begin{frame}\frametitle{A criterion}
A variation of the homotopy colimit criterion:
\begin{tikzcd}[ampersand replacement=\&]
......@@ -720,13 +721,14 @@
be a cocartesian square in $\oo\Cat$. If
\item<2-> $A$,$B$ and $C$ are homologically coherent,
\item<3-> $u$ or $v$ is a folk cofibration,
\item<4-> the square is homotopy cocartesian w.r.t Thomason equivalences,
\pause\pause\pause\pause then $D$ is homologically coherent.
\pause The third condition will usually be the hard one to prove.
\begin{frame}\frametitle{Easy application: homology of globes and spheres}
For every $n\geq 0$, $\sD_n$ is oplax contractible, hence
......@@ -747,8 +749,114 @@
\pause By an immediate induction, $\sS_n$ is homologically coherent
(and has the homotopy type of an $n$\nbd{}sphere).
We would like to understand which 2-categories are homologically
\item<2-> For simplification, we focus on \emph{free} 2-categories.
\item<3-> This boils down to the following: given a cocartesian square
\begin{tikzcd}[ampersand replacement=\&]
\sS_1 \ar[d,"i_1"'] \ar[r] \& P \ar[d] \\
\sD_2 \ar[r] \& P'
\ar[from=1-1,to=2-2,"\ulcorner",very near end,phantom]
with $P$ and $P'$ free $2$\nbd{}categories, when is it homotopy
cocartesian w.r.t the Thomason equivalences ?
\item<4-> I do not have a general answer to this question...
\item<5-> However, using tools that I don't have time to explain, I know how to answer this question in many concrete situations.
\begin{frame}\frametitle{Zoology of 2-categories: basic examples}
For $n, m \geq 0$, let $A_{(m,n)}$ be the free $2$\nbd{}category, with
one generating $2$\nbd{}cell whose source is a chain of length $m$ and target
a chain of length $n$:
A_{(m,n)} = \qquad \underbrace{\overbrace{\begin{tikzcd}[column sep=small, ampersand
replacement=\&] \&\bullet
\ar[r,description,"\cdots",phantom,""{name=A,below}] \& \bullet
\ar[rd] \& \\ \bullet \ar[ru] \ar[rd] \& \& \&\bullet \\ \&\bullet
\ar[r,description,"\cdots",phantom,""{name=B,above}] \& \bullet.
\ar[ru]\ar[from=A,to=B,shorten <= 2em, shorten >=
\pause Examples:
\item<4-> $A_{(1,1)}$ is $\sD_2$.
\item<5-> $A_{(0,0)}$ is the $2$\nbd{}category $B$ from Ara and
Maltsiniotis' counter-example.
\begin{frame}\frametitle{Zoology of $2$\nbd{}categories: basic examples}
If $n+m>0$, the $2$\nbd{}category $A_{(m,n)}$ has the homotopy type of a
point and is homologically coherent.
Else, $A_{(0,0)}$ has the homotopy type of a $K(\mathbb{Z},2)$.
\pause Note: for $m+n=1$, the result was not \emph{a priori} clear.
\pause For example:
A_{(1,0)}= \begin{tikzcd}
\bullet \ar[loop,in=50,out=130,distance=1.5cm,""{name=A,below}]
has many non-trivial $2$\nbd{}cells.
\begin{frame}\frametitle{Zoology of $2$\nbd{}categories: variation of spheres}
% \small
\begin{tabular}{ l || c | c }
$2$\nbd{}category & \good{}? & homotopy type \\ \hline \hline \pause
$\begin{tikzcd}[ampersand replacement=\&]
\bullet \ar[r,bend
right=75,""{name=B,above,pos=9/20},""{name=D,above,pos=11/20}] \& \bullet
\ar[from=C,to=D,bend left,Rightarrow] \ar[from=A,to=B,bend
} & yes & $\sS_2$\\
{ $ \begin{tikzcd}[ampersand replacement=\&]
\bullet \ar[r,bend left=75,""{name=A,below}] \ar[r,bend
right=75,""{name=B,above}] \& \bullet \ar[from=A,to=B,bend
right,Rightarrow] \ar[from=B,to=A,bend
\end{tikzcd}$} & yes & $\sS_2$ \\ \hline \pause {$ \begin{tikzcd}[ampersand replacement=\&] \bullet
\ar[r,""{name=A,above}] \& \bullet \ar[from=A,to=A,loop, in=130,
out=50,distance=1cm, Rightarrow] \end{tikzcd}$} & yes &$\sS_2$ \\
\bullet \ar[loop,in=30,out=150,distance=2cm,""{name=A,below}]
\ar[from=A,to=1-1,bend right,Rightarrow]
\ar[from=A,to=1-1,bend left,Rightarrow]
\end{tikzcd}$ } & yes & $\sS_2$ \\ \hline \pause { $\begin{tikzcd} \bullet
\ar[from=A,to=1-1,bend right,Rightarrow] \ar[from=1-1,to=A,bend
\end{tikzcd}$ } & no & $K(\mathbb{Z},2)$ \\ \hline \pause {$\begin{tikzcd}
\bullet \ar[loop,in=120,out=60,distance=1.2cm,Rightarrow]
\end{tikzcd}$} & no & $K(\mathbb{Z},2)$ \\ \hline
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