Commit 6c252796 by Leonard Guetta

### recompiled because weird things going on

parent f6ae46a7
 ... @@ -649,7 +649,7 @@ We now recall an important theorem due to Thomason. ... @@ -649,7 +649,7 @@ We now recall an important theorem due to Thomason. \begin{remark} \begin{remark} It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}categories), we have $\hocolim^{\Th}_{a \in A} (X/a) \simeq X.$ However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation. It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}categories), we have $\hocolim^{\Th}_{a \in A} (X/a) \simeq X.$ However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation. \end{remark} \end{remark} Putting all the pieces together, we are now able to prove the awaited tyheorem. Putting all the pieces together, we are now able to prove the awaited theorem. \begin{theorem}\label{thm:categoriesaregood} \begin{theorem}\label{thm:categoriesaregood} Every $1$\nbd{}category is \good{}. Every $1$\nbd{}category is \good{}. \end{theorem} \end{theorem} ... ...
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