Commit 05c221bd authored by Leonard Guetta's avatar Leonard Guetta
Browse files

dodo

parent 3b3bcd1e
......@@ -449,9 +449,29 @@
This is what I tried to answer in my PhD.
\end{frame}
\begin{frame}
\frametitle{Another point of view on singular homology}
\frametitle{Equivalence of $\oo$\nbd{}categories vs Thomason
equivalences}
\begin{block}{Important Lemma}
Every equivalence of $\oo$\nbd{}categories is a Thomason equivalence.
\end{block}
\pause Consequence: the identity functor $\mathrm{id} : \oo\Cat \to
\oo\Cat$ induces a functor
\[
\mathcal{J} : \Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th}).
\]
\pause
\underline{Remark}: The converse of the above lemma is false. For
example
\[
\sD_1 \to \sD_0
\]
is a Thomason equivalence but not an equivalence of $\oo$\nbd{}categories.
\end{frame}
\begin{frame}
\frametitle{Singular homology as a derived functor}
\begin{alertblock}{Theorem (Guetta - 2020)}
The functor $\lambda : \oo\Cat \to \Ch$ is left derivable w.r.t to the
The functor $\lambda : \oo\Cat \to \Ch$ is left derivable w.r.t the
\emph{Thomason equivalences} on $\oo\Cat$ and we have
\[
\sH^{\sing}\simeq \LL \lambda : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).
......@@ -459,8 +479,118 @@
\end{alertblock}
\pause
Hence, both $\sH^{\pol}$ and $\sH^{\sing}$ are obtained as left derived
functors of $\lambda$ but not w.r.t the same class of weak equivalences.
\end{frame}
functors of $\lambda$ but not w.r.t the same class of weak equivalences.
\begin{exampleblock}{Corollary}
There is a canonical natural transformation
\[
\begin{tikzcd}[ampersand replacement=\&]
\Ho(\oo\Cat^{\folk})\ar[d,"\mathcal{J}"'] \ar[dr,"\sH^{\pol}",""{name=A,below}]\& \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"']\& \Ho(\Ch).
\ar[from=2-1,to=A,Rightarrow,"\pi"]
\end{tikzcd}
\]
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle{Homologically coherent $\oo$\nbd{}categories}
In other words, for every $\oo$\nbd{}category $C$ we have a map
\[
\pi_C : \sH^{\sing}(C) \to \sH^{\folk}(C),
\]
which is natural in $C$. We refer to it as the \alert{canonical
comparison map}.
\begin{block}{Definition}
An $\oo$\nbd{}category $C$ is \alert{homogically coherent} if the
map
\[
\pi_C : \sH^{\sing}(C) \to \sH^{\folk}(C)
\]
is an isomorphism.
\end{block}
\pause
Goal: Understand which $\oo$\nbd{}categories are homogically coherent.
\end{frame}
\begin{frame}
\frametitle{Polygraphic homology is not homotopical}
Another formal consequence of $\sH^{\sing}$ being left derived of the
abelianization is:
\begin{block}{Proposition}
There exists at least one Thomason equivalence $u : C \to D$ such
that the induced morphism
\[
\sH^{\pol}(C) \to \sH^{\pol}(D)
\]
is \emph{not} an isomorphism.
\end{block}
\pause In other words, if we think of $\oo$\nbd{}categories as models
for homotopy types, then the polygraphic homology is \emph{not} a
well-defined invariant!
\pause
\begin{exampleblock}{New slogan}
The polygraphic homology is a
way of computing the singular homology of homogically coherent
$\oo$\nbd{}categories.
\end{exampleblock}
\end{frame}
\begin{frame}\frametitle{Equivalence of homology in low dimension}
\begin{block}{Proposition}
Let $C$ be \emph{any} $\oo$\nbd{}category. The canonical comparison
map induces an isomorphism
\[\sH^{\sing}_k(C) \to \sH^{\pol}_k(C)\]
for $k=0,1$.
\end{block}
\pause
For all $k\geq 4$, it is possible to find a $C$ such that
\[
H^{\pol}_k(C)\not \simeq H_k^{\sing}(C).
\]
\pause
\begin{exampleblock}{Open question:}
Do we have
\[
H^{\pol}_k(C)\simeq H^{\sing}_k(C)
\]
for $k=2,3$, for any $\oo$\nbd{}category $C$ ?
\end{exampleblock}
\end{frame}
\begin{frame}\frametitle{An abstract criterion}
Back on the triangle:
\[
\begin{tikzcd}[ampersand replacement=\&]
\Ho(\oo\Cat^{\folk})\ar[d,"\mathcal{J}"'] \ar[dr,"\sH^{\pol}",""{name=A,below}]\& \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"']\& \Ho(\Ch).
\ar[from=2-1,to=A,Rightarrow,"\pi"]
\end{tikzcd}
\]
\pause \begin{exampleblock}{Fundamental observation:} $\sH^{\pol}$ and $\sH^{\sing}$ preserve
homotopy colimits but $\J$ does \emph{not} in general.
\end{exampleblock}
% (Because this
% would imply that the canonical comparison map is always an isomorphism.)
\pause
In other words, for a diagram $d : I \to \oo\Cat$, the
canonical map
\[
\hocolim_{I}^{\folk}(d) \to \hocolim_{I}^{\Th}(d)
\]
is not an isomorphism in general.
\pause
Idea: exploit that sometimes it \emph{is} an isomorphism.
\end{frame}
\begin{frame}\frametitle{An abstract criterion}
\begin{block}{Proposition}
Let $C$ be an $\oo$\nbd{}category. Suppose that there exists $d : I
\to \oo\Cat$ such that:
% \begin{enumerate}[label=($\roman$)]
% \item \[
% \hocolim^{\pol}_I(d)\simeq \hocolim^{\Th}_I(d) \simeq C
% \]
% \end{enumerate}
\end{block}
\end{frame}
\end{document}
%%% Local Variables:
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment