Commit d70ea50d by Leonard Guetta

### intro presque terminée

parent 8163b114
 ... @@ -320,7 +320,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c ... @@ -320,7 +320,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c \mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\lambda_n(F)}{\longrightarrow} \lambda_n(C') \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma'_n.\qedhere \mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\lambda_n(F)}{\longrightarrow} \lambda_n(C') \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma'_n.\qedhere \] \] \end{proof} \end{proof} \section{Polygraphic homology} \section{Polygraphic homology}\label{section:polygraphichmlgy} In the next Lemma, recall the definition of \emph{homotopy of chain complexes} (for example from \cite[Definition 1.4.4]{weibel1995introduction} where it is called \emph{chain homotopy}). In the next Lemma, recall the definition of \emph{homotopy of chain complexes} (for example from \cite[Definition 1.4.4]{weibel1995introduction} where it is called \emph{chain homotopy}). \begin{lemma}\label{lemma:abeloplax} \begin{lemma}\label{lemma:abeloplax} Let $u, v : C \to D$ be two $\oo$-functors. If there is an oplax transformation $\alpha : u \Rightarrow v$, then there is a homotopy of chain complexes from $\lambda(u)$ to $\lambda(v)$. Let $u, v : C \to D$ be two $\oo$-functors. If there is an oplax transformation $\alpha : u \Rightarrow v$, then there is a homotopy of chain complexes from $\lambda(u)$ to $\lambda(v)$. ... @@ -469,7 +469,7 @@ The following proposition is an immediate consequence of the previous lemma. ... @@ -469,7 +469,7 @@ The following proposition is an immediate consequence of the previous lemma. is an isomorphism. is an isomorphism. \end{proposition} \end{proposition} \begin{paragr}\label{paragr:polhmlgythomeq} \begin{paragr}\label{paragr:polhmlgythomeq} Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder if it is true that \emph{any} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later, it is not the case. Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder whether it is true that \emph{any} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later, it is not the case. \end{paragr} \end{paragr} \begin{remark} \begin{remark} Proposition \ref{prop:oplaxhmtpypolhmlgy} is also true if we replace oplax'' by lax''. Proposition \ref{prop:oplaxhmtpypolhmlgy} is also true if we replace oplax'' by lax''. ... @@ -497,7 +497,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins ... @@ -497,7 +497,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins \] \] is homotopy cocontinuous. is homotopy cocontinuous. \end{proposition} \end{proposition} \section{Singular homology as derived abelianization} \section{Singular homology as derived abelianization}\label{section:singhmlgyderived} We have seen in the previous section that the polygraphic homology functor is the total left derived functor of $\lambda : \oo\Cat \to \Ch$ when $\oo\Cat$ is equipped with folk weak equivalences. As it turns out, the abelianization functor is also totally left derivable when $\oo\Cat$ is equipped with Thomason equivalences and the total left derived functor is the Singular homology functor. In order to prove this result, we first need a few technical lemmas. We have seen in the previous section that the polygraphic homology functor is the total left derived functor of $\lambda : \oo\Cat \to \Ch$ when $\oo\Cat$ is equipped with folk weak equivalences. As it turns out, the abelianization functor is also totally left derivable when $\oo\Cat$ is equipped with Thomason equivalences and the total left derived functor is the Singular homology functor. In order to prove this result, we first need a few technical 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 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 equivalences. ... ...
 ... @@ -221,15 +221,24 @@ The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is ... @@ -221,15 +221,24 @@ The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is $\oo$\nbd{}categories, taking a weak polygraphic resolution'' of a strict $\oo$\nbd{}categories, taking a weak polygraphic resolution'' of a strict $\oo$\nbd{}category is not the same as taking a polygraphic resolution. In $\oo$\nbd{}category is not the same as taking a polygraphic resolution. In fact, when trying to compute the weak polygraphic homology of $B$, it seems that it gives the fact, when trying to compute the weak polygraphic homology of $B$, it seems that it gives the homology groups of a $K(\mathbb{Z},2)$-space. From this observation, it is natural to wonder homology groups of a $K(\mathbb{Z},2)$-space, which is what we would have expected of the polygraphic homology in the first place. From this observation, it is tempting to make the following conjecture: whether the fact that polygraphic and singular homologies of strict $\oo$\nbd{}categories do not coincide is a defect due to working in too narrow a setting. It is then tempting to make the following conjecture: \begin{center} \begin{center} The weak polygraphic homology of a (strict) $\oo$\nbd{}category coincide The weak polygraphic homology of a (strict) $\oo$\nbd{}category coincide with it singular homology. with its singular homology. \end{center} \end{center} In other words, we conjecture that the fact that polygraphic and singular homologies of strict $\oo$\nbd{}categories do not coincide is a defect due to working in too narrow a setting. The good'' definition of polygraphic homology ought to be the weak one. We can go even further and conjecture the same thing for weak $\oo$\nbd{}categories. In order to do that, we need a definition of singular homology for weak $\oo$\nbd{}categories. This is conjecturally done as follows. To every weak $\oo$\nbd{}category $C$, one can associates a weak $\oo$\nbd{}groupoid $L(C)$ by formally inverting all the cells of $C$. Then, if we believe in Grothendieck's conjecture (see \cite[Section 2]{maltsiniotis2010grothendieck}), the category of weak $\oo$\nbd{}groupoids equipped with weak equivalences of weak $\oo$\nbd{}groupoids (see Paragraph 2.2 of op.\ cit.) is a model for the homotopy theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has homology groups and we can define the singular homology groups of a weak $\oo$\nbd{}category $C$ as the homology groups of $L(C)$. %% This defines a functor %% $%% L : \mathbf{W}\oo\Cat \to \mathbf{W}\oo\Grpd %%$ \end{named} \end{named} \begin{named}[Organization of the thesis] \begin{named}[Organization of the thesis] In the first chapter, we review some aspects of the theory of $\oo$\nbd{}categories. In particular, In the first chapter, we review some aspects of the theory of $\oo$\nbd{}categories. In particular, ... @@ -241,7 +250,21 @@ The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is ... @@ -241,7 +250,21 @@ The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is if $D$ is free, then so is $C$. The proof of this theorem is long and technical and is broke down if $D$ is free, then so is $C$. The proof of this theorem is long and technical and is broke down into several distinct parts. into several distinct parts. The second chapter is devoted to The second chapter is devoted to recalling some tools of homotopical algebra. More precisely, we quickly present basic aspects of the theory of homotopy colimits using the formalism of Grothendieck's derivators. Note that this chapter does \emph{not} contain any original result and can be skipped at first reading. It was only intended to give the reader a summary of useful results needed which are used in the rest of the dissertation. In the third chapter, we dive into the homotopy theory of $\oo$\nbd{}categories. It is there that we define the different notions of weak equivalences for $\oo$\nbd{}categories and compare them. The two most significant new results to be found in this chapter are probably Proposition \ref{prop:folkisthom}, which states that every equivalence of $\oo$\nbd{}categories is a Thomason equivalence, and Proposition \ref{prop:folkthmA}, which states that equivalences of $\oo$\nbd{}categories satisfy a property reminiscing of Quillen's Theorem $A$ \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theorem,ara2020theoreme}. The fourth chapter is certainly the most important of the dissertation as it is there that we define the polygraphic and singular homologies of $\oo$\nbd{}categories and properly formulate the problem of their comparison. Up to Section \ref{section:polygraphichmlgy} included, all the results were known prior to this thesis (at least in the folklore), but starting from Section \ref{section:singhmlgyderived} all the results are original. \end{named} \end{named} %% Let us come back to the canonical $2$-triangle %% Let us come back to the canonical $2$-triangle %% \[ %% \[ ... ...
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 ... @@ -104,6 +104,7 @@ headpunct=.] ... @@ -104,6 +104,7 @@ headpunct=.] % oo-categories % oo-categories \newcommand{\oo}{\omega} \newcommand{\oo}{\omega} \newcommand{\Cat}{\mathbf{Cat}} \newcommand{\Cat}{\mathbf{Cat}} \newcommand{\Grpd}{\mathbf{Grpd}} \newcommand{\ooCat}{\mathbf{\oo Cat}} \newcommand{\ooCat}{\mathbf{\oo Cat}} \newcommand{\nCat}{n \mathbf{Cat}} \newcommand{\nCat}{n \mathbf{Cat}} \newcommand{\ooGrph}{\mathbf{\oo Grph}} \newcommand{\ooGrph}{\mathbf{\oo Grph}} ... ...
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