Commit 48fd8f1e by Leonard Guetta

### security commit

parent d565932a
 ... @@ -771,3 +771,5 @@ For any $n \geq 0$, consider the following cocartesian square ... @@ -771,3 +771,5 @@ For any $n \geq 0$, consider the following cocartesian square \begin{proposition} \begin{proposition} Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or $n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$. Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or $n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$. \end{proposition} \end{proposition} \section{Zoology of $2$-categories: A stub of a criterion} \section{Bubble-free'' $2$-categories}
 ... @@ -56,7 +56,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{ ... @@ -56,7 +56,7 @@ In particular, this means that we have a morphism of localizers$\kappa : (\Psh{ For an $\oo$-category $X$, $\sH(X)$ is the \emph{homology of $X$}. For an $\oo$-category $X$, $\sH(X)$ is the \emph{homology of $X$}. \end{definition} \end{definition} \begin{paragr} \begin{paragr} In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{true'' homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later. In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later. Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space. Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space. \end{paragr} \end{paragr} \begin{remark} \begin{remark} ... @@ -214,8 +214,8 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{ ... @@ -214,8 +214,8 @@ In particular, this means that we have a morphism of localizers$\kappa : (\Psh{ From Theorem \ref{thm:cisinskiII}, this morphism of op-prederivators is cocontinuous. This property will be extremely important in the sequel. From Theorem \ref{thm:cisinskiII}, this morphism of op-prederivators is cocontinuous. This property will be extremely important in the sequel. \end{paragr} \end{paragr} \todo{Expliquer concrètement comment calculer l'homologie polygraphique ?} \todo{Expliquer concrètement comment calculer l'homologie polygraphique ?} \section{True'' homology as derived abelianization} \section{Street 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. 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 Street homology functor for $\oo$-categories. In order to prove this result, we need a few lemmas. \begin{lemma}\label{lemma:nuhomotopical} \begin{lemma}\label{lemma:nuhomotopical} Let $\nu : \Ch \to \oo\Cat$ be the right adjoint to the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason weak equivalences. Let $\nu : \Ch \to \oo\Cat$ be the 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} \end{lemma} ... @@ -250,7 +250,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{ ... @@ -250,7 +250,7 @@ In particular, this means that we have a morphism of localizers$\kappa : (\Psh{ $\[ \LL \lambda : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch) \LL \lambda : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)$ \] is isomorphic to the true'' homology is isomorphic to the Street homology $\[ \sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch). \sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).$ \] ... @@ -293,7 +293,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{ ... @@ -293,7 +293,7 @@ In particular, this means that we have a morphism of localizers$\kappa : (\Psh{ \overline{\kappa} \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \ar[r,shift left]& \ar[l,shift left] \Ho(\Ch) : M \overline{N_{\oo}} \overline{\nu} \simeq \overline{\nu}. \overline{\kappa} \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \ar[r,shift left]& \ar[l,shift left] \Ho(\Ch) : M \overline{N_{\oo}} \overline{\nu} \simeq \overline{\nu}. \end{tikzcd} \end{tikzcd} \] \] From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $\LL^{\Th} \lambda \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the true'' homology. From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $\LL^{\Th} \lambda \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the Street homology. \end{proof} \end{proof} \begin{remark} \begin{remark} Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ nor $\lambda : \oo\Cat \to \Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does. Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ nor $\lambda : \oo\Cat \to \Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does. ... @@ -437,7 +437,7 @@ We shall now proceed to give an abstract criterion to find \good{} $\oo$-categor ... @@ -437,7 +437,7 @@ We shall now proceed to give an abstract criterion to find \good{} $\oo$-categor $\[ \sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch) \sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)$ \] and the true'' homology and the Street homology $\[ \sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch) \sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)$ \] ... @@ -496,3 +496,5 @@ The previous proposition admits the following corollary, which will be of great ... @@ -496,3 +496,5 @@ The previous proposition admits the following corollary, which will be of great \begin{proof} \begin{proof} The fact that $A$,$B$ and $C$ are free and one of the morphism $u$ or $f$ is a folk cofibration ensure that the square is folk homotopy cocartesian. The conclusion follows then from Proposition \ref{prop:criteriongoodcat}. The fact that $A$,$B$ and $C$ are free and one of the morphism $u$ or $f$ is a folk cofibration ensure that the square is folk homotopy cocartesian. The conclusion follows then from Proposition \ref{prop:criteriongoodcat}. \end{proof} \end{proof} \section{Polygraphic homology and truncation} \section{Homology and Homotopy of $\oo$-categories in low dimension}
 ... @@ -2,6 +2,12 @@ ... @@ -2,6 +2,12 @@ \usepackage{mystyle} \usepackage{mystyle} %%% Watermark \usepackage{draftwatermark} \SetWatermarkText{DRAFT} \SetWatermarkScale{2} %%% \title{Homology of strict $\omega$-categories} \title{Homology of strict $\omega$-categories} \author{Léonard Guetta} \author{Léonard Guetta} \begin{document} \begin{document} ... @@ -9,7 +15,7 @@ ... @@ -9,7 +15,7 @@ \maketitle \maketitle \tableofcontents \tableofcontents \abstract{In this dissertation, we study the homology of strict $\oo$-categories. More precisely, we intend to compare the classical'' homology of an $\oo$-category (defined as the homology of its Street nerve) with its polygraphic homology. Along the way, we prove several important result concerning free strict $\oo$\nbd-categories over polygraphs-or-computad and concerning the homotopy theory of strict $\oo$\nbd-categories. } \include{omegacat} \include{omegacat} \include{homtheo} \include{homtheo} \include{hmtpy} \include{hmtpy} ... ...
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