Commit 05c221bd by Leonard Guetta

### 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: ... ...
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