Commit 6a5499bb authored by Leonard Guetta's avatar Leonard Guetta
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......@@ -216,7 +216,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\todo{Expliquer concrètement comment calculer l'homologie polygraphique ?}
\section{``True'' homology as derived abelianization}
We have seen in the previous section that the polygraphic homology is the left derived of the abelianization functor with respect to the canonical weak equivalences on $\oo\Cat$.\todo{Uniformiser appellations} As it turns out, the abelianization functor is also left derivable with respect to the Thomason weak equivalences and the left derived functor is the ``true'' homology functor for $\oo$-categories. In order to prove this result, we need a few lemmas.
\begin{lemma}
\begin{lemma}\label{lemma:nuhomotopical}
Let $\nu : \Ch \to \oo\Cat$ be a right adjoint to the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason weak equivalences.
\end{lemma}
\begin{proof}
......@@ -225,10 +225,40 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\begin{remark}
The proof of the previous lemma shows the stronger result that $\nu$ sends weak equivalences of chain complexes to weak equivalences for the canonical model structure on $\oo\Cat$. This will be of no use in the sequel.
\end{remark}
Recall now that the notion of adjunction is valid in any $2$-category. \todo{Peut-être que je l'aurai déjà dit quelque part ça.} We omit the proof of the following lemma, which is the same as when the ambient $2$-category is the $2$-category of categories.
\begin{lemma}
Let
Recall that $c_{\oo} : : \Psh{\Delta} \to \oo\Cat$ is the left adjoint of the nerve functor $N_{\oo} : \oo\Cat \to \Psh{\Delta}$ (see Paragraph \ref{paragr:nerve}).
\begin{lemma}\label{lemma:abelor}
The triangle of functors
\[
\begin{tikzcd}
\Psh{\Delta} \ar[r,"c_{\oo}"] \ar[dr,"\kappa"']& \oo\Cat\ar[d,"\lambda"]\\
&\Ch
\end{tikzcd}
\]
is commutative (up to a canonical isomorphism).
\end{lemma}
\begin{proof}
All the functors involved are cocontinuous, hence it suffices to prove that the triangle is commutative when pre-composed by the Yoneda embedding $\Delta \to \Psh{\Delta}$. This follows immediately from the description of the orientals in \cite{steiner2004images}.
\end{proof}
Recall now that the notion of adjunction and equivalence is valid in any $2$-category. \todo{Peut-être que je l'aurai déjà dit quelque part ça.} We omit the proof of the following lemma, which is the same as when the ambient $2$-category is the $2$-category of categories.
\todo{Parler d'adjonction dans le chapitre sur les dérivateurs.}
\begin{lemma}\label{lemma:adjeq}
Let $f : x \overset{\longrightarrow}{\longleftarrow} y$ be an adjunction and $h : y \to z$ an equivalence with quasi-inverse $k : z \to y$. Then $hf$ is left adjoint to $gk$.
\end{lemma}
We can now state and prove the promised result.
\begin{proposition}
The abelianization functor $\lambda : \oo\Cat \to $
The abelianization functor $\lambda : \oo\Cat \to \Ch$ is strongly left derivable when $\oo\Cat$ is equipped with the Thomason weak equivalences. Moreover, the left derived morphism
\[
\LL \lambda : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
\]
is isomorphic to the ``true'' homology
\[
\sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).
\]
\end{proposition}
\begin{proof}
It follows from Lemma \ref{lemma:nuhomotopical} that we have a morphism of localizers
\[
(\Ch,\W_{\Ch}) \to (\oo\Cat,\W^{\Th})
\]
\todo{Uniformiser les notations pour les é.f des cmplxs de chaines}. This
\end{proof}
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