Commit 3b3bcd1e authored by Leonard Guetta's avatar Leonard Guetta
Browse files

slowly but surely

parent e7cb9aa8
......@@ -324,7 +324,7 @@
\frametitle{Equivalence of $\oo$\nbd{}categories and the folk model
\begin{alertblock}{Theorem (Lafont,Métayer,Worytkiewicz - 2009)}
\begin{alertblock}{Theorem (Lafont,Métayer,Worytkiewicz - 2010)}
There exists a model structure on $\oo\Cat$ such that:
\item the weak equivalences are the equivalences of
......@@ -380,7 +380,7 @@
\frametitle{Polygraphic homology}
\begin{alertblock}{Proposition (folklore ?)}
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 $\Ch$.
......@@ -400,17 +400,69 @@
% \frametitle{}
\frametitle{Polygraphic homology vs singular homology}
A natural question:
Let $C$ be an $\oo$\nbd{}category. Do we have $\sH^{\pol}(C) \simeq
\sH^{\sing}(C)$ ?
% \pause
% Answer : In general, \textbf{no} !
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).
However, there are $\oo$\nbd{}categories $C$ for which
$\sH^{\pol}(C) \not \simeq \sH^{\sing}(C)$.
\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 $k$.
% More generally, we can construct for every $n\geq
% 2$ an $n$\nbd{}category $C$ for which $\sH^{\pol}(C)\not \simeq \sH^{\sing}(C)$.
% \frametitle{The big question}
\begin{exampleblock}{The fundamental question}
For which $\oo$\nbd{}categories $C$ do we have $\sH^{\pol}(C)\simeq
\sH^{\sing}(C)$ ?
This is what I tried to answer in my PhD.
\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).
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.
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