Commit 3b3bcd1e by Leonard Guetta

### slowly but surely

parent e7cb9aa8
 ... ... @@ -324,7 +324,7 @@ \begin{frame} \frametitle{Equivalence of $\oo$\nbd{}categories and the folk model structure} \begin{alertblock}{Theorem (Lafont,Métayer,Worytkiewicz - 2009)} \begin{alertblock}{Theorem (Lafont,Métayer,Worytkiewicz - 2010)} There exists a model structure on $\oo\Cat$ such that: \begin{itemize}[label=$\bullet$] \item the weak equivalences are the equivalences of ... ... @@ -380,7 +380,7 @@ \end{frame} \begin{frame} \frametitle{Polygraphic homology} \begin{alertblock}{Proposition (folklore ?)} \begin{alertblock}{Proposition} The functor $\lambda : \oo\Cat \to \Ch$ is left Quillen w.r.t the folk model structure on $\oo\Cat$ and the projective model structure on $\Ch$. \end{alertblock} ... ... @@ -400,17 +400,69 @@ TODO \end{frame} \begin{frame} % \frametitle{} \frametitle{Polygraphic homology vs singular homology} A natural question: \begin{center} Let $C$ be an $\oo$\nbd{}category. Do we have $\sH^{\pol}(C) \simeq \sH^{\sing}(C)$ ? \end{center} % \pause % Answer : In general, \textbf{no} ! \pause A first partial answer: \begin{block}{Theorem (Lafont, Métayer - 2009)} For every monoid $M$ (considered as an $\oo$\nbd{}category), we have $\sH^{\pol}(M) \simeq \sH^{\sing}(M).$ \end{block} \pause However, there are $\oo$\nbd{}categories $C$ for which $\sH^{\pol}(C) \not \simeq \sH^{\sing}(C)$. \end{frame} \begin{frame} \frametitle{Ara and Maltsiniotis' counter-example} Let $B$ be the commutative monoid $(\mathbb{N},+)$ considered as a $2$\nbd{}category with exactly one $0$\nbd{}cell and one $1$\nbd{}cell: $B = \begin{tikzcd}[ampersand replacement=\&] \bullet \& \bullet \ar[l,shift right] \ar[l,shift left] \& \mathbb{N} \ar[l,shift right] \ar[l,shift left].\end{tikzcd}$ \pause $B$ is free as an $\oo$\nbd{}category and we have $H_k^{\pol}(B)\simeq \begin{cases}\mathbb{Z} &\text{ if } k=0,2 \\ 0 & \text{ otherwise.} \end{cases}$ \pause But (the nerve) of $B$ has the homotopy type of a $K(\mathbb{Z},2)$, hence $H^{\sing}_k(B)$ is non-trivial for \alert{all} even values of $k$. \pause % More generally, we can construct for every $n\geq % 2$ an $n$\nbd{}category $C$ for which $\sH^{\pol}(C)\not \simeq \sH^{\sing}(C)$. \end{frame} \begin{frame} % \frametitle{The big question} \begin{exampleblock}{The fundamental question} For which $\oo$\nbd{}categories $C$ do we have $\sH^{\pol}(C)\simeq \sH^{\sing}(C)$ ? \end{exampleblock} \pause This is what I tried to answer in my PhD. \end{frame} \begin{frame} \frametitle{Another point of view on singular homology} \begin{alertblock}{Theorem (Guetta - 2020)} The functor $\lambda : \oo\Cat \to \Ch$ is left derivable w.r.t to the \emph{Thomason equivalences} on $\oo\Cat$ and we have $\sH^{\sing}\simeq \LL \lambda : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).$ \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} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t ... ...
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