Commit b9ad44ff authored by Leonard Guetta's avatar Leonard Guetta
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ouf ! Plus que l'intro et de la relecture. Je devrais changer la notion H_k...

ouf ! Plus que l'intro et de la relecture. Je devrais changer la notion H_k dans le dernier chapitre car il y a une ambiguité avec la notation de l'homologie
parent 6b618234
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......@@ -1435,7 +1435,7 @@ Finally, we obtain the result we were aiming for.
\]
From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha^{\sing}$ induces isomorphisms \[H_k^{\sing}(C) \simeq H_k(\lambda(C))\] for $k \in \{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha^{\pol}$ induces isomorphisms $H_k^{\pol}(C) \simeq H_k(\lambda(C))$ for $k \in \{0,1\}$. The result follows then from an immediate 2-out-of-3 property.
\end{proof}
\begin{paragr}
\begin{paragr}\label{paragr:conjectureH2}
A natural question following the above proposition is:
\begin{center}
For which $k \geq 0$ do we have $H_k^{\sing}(C) \simeq H_k^{\pol}(C)$ for every $\oo$\nbd{}category $C$ ?
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......@@ -784,3 +784,17 @@ In this vast generalization of category theory, the objects of study have object
& \ho(\Ch)\ar[from=1-2,to=A,"\pi",Rightarrow]
\end{tikzcd}
\]
%%%% CLASSIFYING BUBBLES
In fact, this $2$\nbd{}category classifies bubbles in the
sense that the functor
\begin{align*}
2\Cat &\to \Set \\
C &\mapsto \Hom_{2\Cat}(B^2\mathbb{N},C)
\end{align*}
is canonically isomorphic to the functor that sends a $2$\nbd{}category to its
set of bubbles.
\todo{À finir}
\ No newline at end of file
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