### edited typos and did some small corrections suggested by Garner

parent beb43c6a
 ... ... @@ -312,20 +312,17 @@ From the previous proposition, we deduce the following very useful corollary. is quasi-injective on arrows; the remaining case being symmetric. Let use denote by $E$ the set of objects of $B$ that are in the image of $\beta$. For each element $x$ of $E$, we denote by $F_x$ the fiber'' of $x$, that is the set of objects of $A$ that $\beta$ sends to $x$. We consider the set $E$ and each $F_x$ as discrete reflexive graphs, i.e.\ reflexive graphs $\beta$. We consider this set as well as the set $A_0$ of objects of $A$ as discrete reflexive graphs, i.e.\ reflexive graphs with no non-unit arrows. Now, let $G$ be the reflexive graph defined by the following cocartesian square $\begin{tikzcd} \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\ A_0\ar[r] \ar[d] & E \ar[d]\\ A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}$ where the morphism $\coprod_{x \in E}F_x \to A$ is induced by the inclusion of each $F_x$ in $A$, and the morphism $\coprod_{x \in E}F_x \to E$ is the only one that sends an element $a \in F_x$ to $x$. In other words, $G$ is where the morphism $A_0 \to A$ is the canonical inclusion, and the morphism $A_0 \to E$ is induced by the restriction of $\beta$ on objects. In other words, $G$ is obtained from $A$ by collapsing the objects that are identified through $\beta$. It admits the following explicit description: $G_0$ is (isomorphic to) $E$ and the set of non-unit arrows of $G$ is (isomorphic to) the set of ... ... @@ -337,7 +334,7 @@ From the previous proposition, we deduce the following very useful corollary. Now, we have the following solid arrow commutative diagram $\begin{tikzcd} \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\ A_0 \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\ A \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\ &&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd} ... ... @@ -351,7 +348,7 @@ From the previous proposition, we deduce the following very useful corollary. obtained earlier, we obtain a diagram of three cocartesian squares: \[ \begin{tikzcd}[row sep = large] \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]&\\ A_0\ar[r] \ar[d] & E \ar[d]&\\ A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\ C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description] ... ... @@ -368,15 +365,19 @@ From the previous proposition, we deduce the following very useful corollary. L of square \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Hence, in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is that the image by L of square \textcircled{\tiny \textbf{2}} is homotopy cocartesian. On the other hand, the morphism cocartesian. On the other hand, the morphisms \[ \coprod_{x \in E}F_x \to A A_0 \to A$ is a monomorphism and thus, using Corollary and $A_0 \to C$ are monomorphisms and thus, using Corollary \ref{cor:hmtpysquaregraph}, we deduce that the image by $L$ of square \textcircled{\tiny \textbf{1}} and of the pasting of squares \textcircled{\tiny \textbf{1}} and \textcircled{\tiny \textbf{2}} are homotopy cocartesian. This proves that the image by $L$ of are homotopy cocartesian. By Lemma \ref{lemma:pastinghmtpycocartesian} again, this proves that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy cocartesian. \end{proof} We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition ... ... @@ -1105,7 +1106,7 @@ of $2$-categories. \begin{proof} Let $r : \Delta_n \to \Delta_1$ the unique functor such that $r(n)=1\text{ and } r(k)=1 \text{ for } k\neq n. r(0)=0\text{ and } r(k)=1 \text{ for } k>0.$ By definition we have $r \circ i = 1_{\Delta_1}$. Now, the natural order on $\Delta_n$ induces a natural transformation ... ... @@ -1217,12 +1218,12 @@ results at the end of the previous section. \begin{equation}\label{square:lemniscate} \begin{tikzcd} \sD_0 \ar[r] \ar[d] & A_{(1,0)} \ar[d]\\ A_{(0,1)} \ar[r] & P. \ar[from=1-1,to=2-2,very near A_{(1,0)} \ar[r] & P. \ar[from=1-1,to=2-2,very near end,"\ulcorner",phantom] \end{tikzcd} \end{equation} Since $\sD_0$, $A_{(1,0)}$ and $A_{(0,1)}$ are all free and \good{} and since $\sD_0 \to A_{(1,0)}$ and $\sD_0 \to A_{(0,1)}$ are folk cofibrations, all we Since $\sD_0$, $A_{(1,0)}$ are free and \good{} and since $\sD_0 \to A_{(1,0)}$ is a folk cofibration, all we have to show to prove that $P$ is \good{} is that the above square is Thomason homotopy cocartesian. Notice that the $2$\nbd{}category $A_{(1,0)}$ is obtained as the following amalgamated sum ... ... @@ -1252,7 +1253,7 @@ Let us now get into more sophisticated examples. \item generating $2$\nbd{}cells: $\alpha : f \Rightarrow g$, $\beta: g \Rightarrow f$. \end{itemize} In picture, this gives In pictures, this gives $\begin{tikzcd} A \ar[r,bend left=75,"f",""{name=A,below}] \ar[r,bend ... ... @@ -1287,13 +1288,27 @@ Let us now get into more sophisticated examples. \item F(\alpha)=\gamma and F(\beta)=1_h. \end{itemize} We wish to prove that this 2\nbd{}functor is a Thomason equivalence. Since it is an isomorphism on objects, it suffices to prove that the functor induced it is an isomorphism on objects, it suffices to prove that the functors induced by F \[ F_{A,A} : P(A,A) \to P'(F(A),F(A)),$ $F_{B,B} : P(B,B) \to P'(F(B),F(B)),$ $F_{B,A} : P(B,A) \to P'(F(B),F(A))$ and $F_{A,B} : P(A,B) \to P'(F(A),F(B))$ is a Thomason equivalence of categories (Corollary \ref{cor:criterionThomeqI}). The category $P(A,B)$ is the free category on the are Thomason equivalences of categories (Corollary \ref{cor:criterionThomeqI}). For the first two ones, this follows trivially from the fact that the categories $P(A,A)$, $P'(A',A')$, $P(B,B)$ and $P'(B',B')$ are all isomorphic to $\sD_0$. For the third one, this follows trivially from the fact that the categories $P(B,A)$ and $P'(B',A')$ are the empty category. For the fourth one, this can be seen as follows. The category $P(A,B)$ is the free category on the graph $\begin{tikzcd} ... ... @@ -1313,7 +1328,7 @@ Let us now get into more sophisticated examples. \ref{example:killinggenerator}). In particular, the square \[ \begin{tikzcd} \sD_1 \ar[r,"\langle \beta \rangle"] \ar[d]& P(A,B) \ar[d,"F_{(A,B)}"] \\ \sD_1 \ar[r,"\langle \beta \rangle"] \ar[d]& P(A,B) \ar[d,"F_{A,B}"] \\ \sD_0 \ar[r,"\langle h \rangle" ] & P'(A',B') \end{tikzcd}$ ... ... @@ -1346,7 +1361,7 @@ Let us now get into more sophisticated examples. \item generating $2$\nbd{}cells: $\lambda : l \Rightarrow 1_{A''}$ and $\mu: l \Rightarrow 1_{A''}$. \end{itemize} In picture, this gives In pictures, this gives $\begin{tikzcd}[column sep=huge] A'' \ar[r,bend left=75,"l",""{name=A'',below}]\ar[r,bend ... ... @@ -1447,7 +1462,7 @@ Let us now get into more sophisticated examples. \item generating 2\nbd{}cells: \alpha : f \Rightarrow 1_A and \beta: 1_A \Rightarrow f. \end{itemize} In picture, this gives In pictures, this gives \[ \begin{tikzcd}[column sep=huge] A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend ... ... @@ -1551,7 +1566,7 @@ Let us now move on to bouquets of spheres. \item generating 1\nbd{}cells: f,g: A \to A, \item generating 2\nbd{}cells: \alpha,\beta : f \Rightarrow g. \end{itemize} In picture, this gives: In pictures, this gives: \[ \begin{tikzcd}[column sep=huge] A \ar[r,bend ... ... @@ -1578,8 +1593,8 @@ Now let \sS_2 be labelled as \begin{equation} \begin{tikzcd} \sS_0 \ar[d,"i_1"] \ar[r,"{\langle C,D \rangle}"] & \sS_2 \ar[d]\\ \sD_1 \ar[r] & P. \sS_0 \ar[d,"p"] \ar[r,"{\langle C,D \rangle}"] & \sS_2 \ar[d]\\ \sD_0 \ar[r] & P. \ar[from=1-1,to=2-2,"\ulcorner",phantom,very near end] \end{tikzcd}\label{eq:squarebouquethybrid} \end{equation} ... ... @@ -1589,12 +1604,13 @@ Now let \sS_2 be labelled as have to show that the induced square of \Cat \begin{equation} \begin{tikzcd} V_k(\sS_0) \ar[r,"{V_k(\langle C,D \rangle)}"] \ar[d,"V_k(i_1)"] & V_k(\sS_2) \ar[d]\\ V_k(\sD_1) \ar[r] & V_k(P) V_k(\sS_0) \ar[r,"{V_k(\langle C,D \rangle)}"] \ar[d,"V_k(p)"] & V_k(\sS_2) \ar[d]\\ V_k(\sD_0) \ar[r] & V_k(P) \end{tikzcd}\label{eq:squarebouquethybridvertical} \end{equation} is Thomason homotopy cocartesian for every k \geq 0. Notice first that we trivially have V_0(\sS_0)\simeq \sS_0 and that V_0(\sS_2) is the free category on the trivially have V_k(\sS_0)\simeq \sS_0 and V_k(\sD_0)\simeq \sD_0 for every k\geq 0 and that V_0(\sS_2) is the free category on the graph \[ \begin{tikzcd} ... ... @@ -1628,7 +1644,9 @@ Now let \sS_2 be labelled as In particular, square \eqref{eq:squarebouquethybridvertical} is again a cocartesian square of identification of two objects of a free category, and thus, it is Thomason homotopy cocartesian. This implies that square \eqref{eq:squarebouquethybrid} is Thomason homotopy cocartesian. Since \sS_0, \sD_1 and \sS_2 are \good{} and since the morphisms \langle C, D \rangle : \sS_0 \to \sS_2 and i_1 : \sS_0 \to \sD_1 are folk cofibrations, this proves that P is \good{} \eqref{eq:squarebouquethybrid} is Thomason homotopy cocartesian. Since \sS_0, \sD_0 and \sS_2 are \good{} and since \langle C, D \rangle : \sS_0 \to \sS_2 is a folk cofibration, this proves that P is \good{} and has the homotopy type of the bouquet of a 1\nbd{}sphere with a 2\nbd{}sphere. \end{paragr} ... ... @@ -1639,7 +1657,7 @@ Now let \sS_2 be labelled as \item generating 1\nbd{}cells: f,g : A \to B, \item generating 2\nbd{}cells: \alpha,\beta,\gamma : f \Rightarrow g. \end{itemize} In picture, this gives In pictures, this gives \[ \begin{tikzcd}[column sep=huge] A \ar[r,bend ... ... @@ -1689,7 +1707,7 @@ Now let \sS_2 be labelled as \item generating 2\nbd{}cells: \alpha,\beta:f \Rightarrow g and \delta,\gamma:g \Rightarrow h. \end{itemize} In picture, this gives: In pictures, this gives: \[ \begin{tikzcd}[column sep=huge] A \ar[r,bend ... ... @@ -1708,7 +1726,7 @@ Now let \sS_2 be labelled as sub-2\nbd{}category of P spanned by A, B, g, h, \gamma and \delta and let P_2 be the sub-2\nbd{}category of P spanned by A, B, f, g, \alpha and \beta. The 2\nbd{}categories P_1 and P_2 are copies of \sS_2, and P_0 is a copy of \sS_1. Moreover, we have a are copies of \sS_2, and P_0 is a copy of \sD_1. Moreover, we have a cocartesian square of inclusions \begin{equation}\label{squarebouquet} \begin{tikzcd} ... ... @@ -1834,18 +1852,23 @@ Now let \sS_2 be labelled as$ This implies that square \eqref{squarebouquetvertical} is cocartesian for $k=0$ and in virtue of Corollary \ref{cor:hmtpysquaregraph} it is also Thomason homotopy cocartesian for this value of $k$. For $k>0$, the category $V_k(P')$ has two objects $A$ and $B$ and an arrow $A \to B$ is a $k$\nbd{}tuple of one of the following form \begin{itemize}[label=-] \item $(1_f,\cdots,1_f,\alpha,1_g,\cdots,1_g)$, \item $(1_f,\cdots,1_f,\beta,1_g,\cdots,1_g)$, \item $(1_f,\cdots,1_f)$, \item $(1_g,\cdots,1_g)$, \end{itemize} and these are the only non-trivial arrows of $V_k(P')$. This means that $V_k(P')$ is the free category on the graph that has two objects and $2k+2$ parallel arrows between these two objects. The same goes for $V_k(P'')$ since $P'$ and $P''$ are isomorphic. Similarly, the category $V_k(P)$ is free on the graph cocartesian for this value of $k$. For $k>0$, since $P'$ and $P''$ are both isomorphic to $\sS_2$, we have already seen in \ref{paragr:variationsphere} that $V_k(P')$ and $V_k(P'')$ are (isomorphic) to the free category on the graph that has two objects and $2k+2$ parallel arrows between these two objects. % the category $V_k(P')$ has two objects $A$ % and $B$ and an arrow $A \to B$ is a $k$\nbd{}tuple of one of the following form % \begin{itemize}[label=-] % \item $(1_f,\cdots,1_f,\alpha,1_g,\cdots,1_g)$, % \item $(1_f,\cdots,1_f,\beta,1_g,\cdots,1_g)$, % \item $(1_f,\cdots,1_f)$, % \item $(1_g,\cdots,1_g)$, % \end{itemize} % and these are the only non-trivial arrows of $V_k(P')$. This means that $V_k(P')$ % is the free category on the graph that has two objects and $2k+2$ parallel % arrows between these two objects. The same goes for $V_k(P'')$ since $P'$ and % $P''$ are isomorphic. Similarly, the category $V_k(P)$ is free on the graph that has three objects $A$, $B$, $C$, whose arrows from $A$ to $B$ are $k$\nbd{}tuples of one of the following form \begin{itemize}[label=-] ... ... @@ -1889,8 +1912,8 @@ homotopy type of the torus. From now on, we will use concatenation instead of the symbol $\comp_0$ for the $0$\nbd{}composition. For example, $fg$ will stand for $f \comp_0 g$. With this notation, the set $1$\nbd{}cells of $P$ is canonically isomorphic to the set of finite words on the alphabet $\{f,g\}$ and the set of $2$\nbd{}cells of $P$ is canonically isomorphic to the set of finite words on the alphabet set of finite words in the alphabet $\{f,g\}$ and the set of $2$\nbd{}cells of $P$ is canonically isomorphic to the set of finite words in the alphabet $\{f,g,\alpha\}$. For a $1$\nbd{}cell $w$ such that $f$ appears $n$ times in $w$ and $g$ appears $m$ times in $w$, it is a simple exercise left to the reader to show that there exists a unique $2$\nbd{}cell of $P$ from $w$ to the ... ... @@ -2026,7 +2049,7 @@ homotopy type of the torus. \end{itemize} \end{definition} \begin{paragr} In picture, a bubble $x$ is represented as In pictures, a bubble $x$ is represented as \[ \begin{tikzcd} A \ar[r,bend left=75,"1_A",""{name=A,below}] \ar[r,bend ... ...
 ... ... @@ -72,7 +72,7 @@ \vfill {\slshape Présentée et soutenue publiquement le 15 janvier 2021 \\ {\slshape Présentée et soutenue publiquement le 28 janvier 2021 \\ devant un jury composé de : \par} \vspace{0.8cm} ... ... @@ -96,26 +96,67 @@ % \includegraphics[width=0.6\textwidth]{license.png} % \vspace*{-3cm} \end{titlepage} \abstract{In this dissertation, we study the homology of strict $\oo$\nbd{}categories. More precisely, we intend to compare the classical'' \end{titlepage} \selectlanguage{french} \begin{abstract} Dans cette thèse, on compare l'homologie \og classique \fg{} d'une $\oo$\nbd{}catégorie (définie comme l'homologie de son nerf de Street) avec son homologie polygraphique. Plus précisément, on prouve que les deux homologies ne coïncident pas en général et qualifions d'\emph{homologiquement cohérente} les $\oo$\nbd{}catégories particulières pour lesquelles l'homologie polygraphique coïncide effectivement avec l'homologie du nerf. Le but poursuivi est de trouver des critères abstraits et concrets permettant de détecter les $\oo$\nbd{}catégories homologiquement cohérentes. Par exemple, on démontre que toutes les (petites) catégories, que l'on considère comme des $\oo$\nbd{}catégories strictes dont toutes les cellules au-delà de la dimension $1$ sont des unités, sont homologiquement cohérente. On introduit également la notion de $2$\nbd{}catégorie \emph{sans bulles} et on conjecture qu'une $2$\nbd{}catégorie cofibrante est homologiquement cohérente si et seulement si elle est sans bulles. On démontre également des résultats importants concernant les $\oo$\nbd{}catégories strictes qui sont libres sur un polygraphique, comme le fait que si $F : C \to D$ est un $\oo$\nbd{}foncteur discret de Conduché et si $D$ est libre sur un polygraphe alors $C$ l'est aussi. Dans son ensemble, cette thèse établit un cadre général dans lequel étudier l'homologie des $\oo$\nbd{}catégories en faisant appels à des outils d'algèbre homotopique abstraite, tels que la théorie des catégorie de modèles de Quillen ou la théorie des dérivateurs de Grothendieck. \end{abstract} \selectlanguage{english} \begin{abstract} In this dissertation, we compare the classical'' homology of an $\oo$\nbd{}category (defined as the homology of its Street nerve) with its polygraphic homology. Along the way, we prove several important results concerning free strict $\oo$\nbd{}categories on polygraphs (also known as computads) and concerning the homotopy theory of strict $\oo$\nbd{}categories. } nerve) with its polygraphic homology. More precisely, we prove that both homologies generally do not coincide and call \emph{homologically coherent} the particular strict $\oo$\nbd{}categories for which polygraphic homology and homology of the nerve do coincide. The goal pursued is to find abstract and concrete criteria to detect homologically coherent $\oo$\nbd{}categories. For example, we prove that all (small) categories, considered as strict $\oo$\nbd{}categories with unit cells above dimension $1$, are homologically coherent. We also introduce the notion of \emph{bubble-free} $2$\nbd{}category and conjecture that a cofibrant $2$\nbd{}category is homologically coherent if and only if it is bubble-free. We also prove important results concerning free strict $\oo$\nbd{}categories on polygraphs (also known as computads), such as the fact that if $F : C \to D$ is a discrete Conduché $\oo$\nbd{}functor and $D$ is a free strict $\oo$\nbd{}category on a polygraph, then so is $C$. Overall, this thesis achieves to build a general framework in which to study the homology of strict $\oo$\nbd{}categories using tools of abstract homotopical algebra such as Quillen's theory of model categories or Grothendieck's theory of derivators. \end{abstract} \tableofcontents \include{remerciements} % \include{introduction} % \include{introduction_fr} % \include{omegacat} % \include{homtheo} % \include{hmtpy} % \include{hmlgy} % \include{contractible} % \include{2cat} %\include{remerciements} \include{introduction} \include{introduction_fr} \include{omegacat} \include{homtheo} \include{hmtpy} \include{hmlgy} \include{contractible} \include{2cat} \addcontentsline{toc}{chapter}{Bibliography} \bibliographystyle{alpha} ... ...
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