### security commit

parent d9419d8c
 ... ... @@ -179,7 +179,7 @@ From the previous proposition, we deduce the following very useful corollary. L(C) \ar[r,"L(\gamma)"]& L(D) \end{tikzcd} \] is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences. is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason equivalences. \end{corollary} \begin{proof} Since the nerve $N$ induces an equivalence of op-prederivators ... ... @@ -229,7 +229,7 @@ From the previous proposition, we deduce the following very useful corollary. L(C) \ar[r,"L(\gamma)"] &L(D) \end{tikzcd} \] is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences. is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason equivalences. \end{proposition} \begin{proof} The case where $\alpha$ or $\beta$ is both injective on objects and quasi-injective on arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is quasi-injective on arrows; the remaining case being symmetric. ... ... @@ -295,7 +295,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \sD_1 \ar[r] & C', \end{tikzcd} \] where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}. where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}. \end{example} \begin{example}[Killing a generator] Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by killing'' $f$, i.e. defined with the following cocartesian square: ... ... @@ -305,7 +305,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \sD_0 \ar[r] & C'. \end{tikzcd} \] Then, this above square is homotopy cocartesion in $\Cat$ (equipped with the Thomason weak equivalences). Indeed, it obviously is the image of a square in $\Rgrph$ by the functor $L$ and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$. Then, this above square is homotopy cocartesian in $\Cat$ (equipped with Thomason equivalences). Indeed, it obviously is the image of a square in $\Rgrph$ by the functor $L$ and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$. \end{example} \begin{remark} Note that in the previous example, we see that it was useful to consider the category of reflexive graphs and not only the category of graphs because the map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs. \todo{À mieux dire ?} ... ... @@ -579,7 +579,7 @@ In the definition of the bisimplicial nerve of a $2$-category we gave, we have p which enables us to compare the homotopy theory of $2\Cat$ with the homotopy theory of bisimplicial sets. \end{paragr} \begin{lemma}\label{lemma:binervthom} A $2$-functor $F : C \to D$ is a Thomason weak equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets. A $2$-functor $F : C \to D$ is a Thomason equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets. \end{lemma} \begin{proof} It follows from what is shown in \cite[Section 2.1 and Theorem 2.7]{bullejos2003geometry} that there is weak equivalence of simplicial sets ... ... @@ -588,7 +588,7 @@ In the definition of the bisimplicial nerve of a $2$-category we gave, we have p \] which is natural in $C$. This implies what we wanted to show. \end{proof} From this lemma, we deduce two useful criteria to detect Thomason weak equivalences of $2$-categories. From this lemma, we deduce two useful criteria to detect Thomason equivalences of $2$-categories. \begin{corollary}\label{cor:criterionThomeqI} Let $F : C \to D$ be a $2$-functor. If \begin{enumerate}[label=\alph*)] ... ... @@ -600,9 +600,9 @@ From this lemma, we deduce two useful criteria to detect Thomason weak equivalen $C(x,y) \to D(F(x),F(y))$ induced by $F$ is a Thomason weak equivalence of $1$-categories, induced by $F$ is a Thomason equivalence of $1$-categories, \end{enumerate} then $F$ is a Thomason weak equivalence of $2$-categories. then $F$ is a Thomason equivalence of $2$-categories. \end{corollary} \begin{proof} By definition, for every $2$-category $C$ and every $m \geq 0$, we have ... ... @@ -613,7 +613,7 @@ From this lemma, we deduce two useful criteria to detect Thomason weak equivalen \end{proof} \begin{corollary} Let $F : C \to D$ be a $2$-functor. If for every $k \geq 0$, $V(F)_k : V(C)_k \to V(D)_k$ is a Thomason weak equivalence of $1$-categories, then $F$ is a Thomason weak equivalence of $2$-categories. $V(F)_k : V(C)_k \to V(D)_k$ is a Thomason equivalence of $1$-categories, then $F$ is a Thomason equivalence of $2$-categories. \end{corollary} \begin{proof} From Lemma \ref{lemma:binervehorizontal}, we now that for every $m \geq 0$, ... ... @@ -744,7 +744,7 @@ For any $n \geq 0$, consider the following cocartesian square \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}, \] where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trival $1$\nbd-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong deformation retract, hence a co-universal Thomason weak equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)} \to A_{(1,n)}$ is also a (co-universal) Thomason weak equivalence and the square is Thomason homotopy cocartesian. Now, the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from Corollary \ref{cor:usefulcriterion} that $A_{(1,n)}$ is \good{}. Finally, since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a point, the fact that the previous square is Thomason homotopy cocartesian implies that $A_{(1,n)}$ has the homotopy type of a point. where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trival $1$\nbd-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong deformation retract, hence a co-universal Thomason equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)} \to A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is Thomason homotopy cocartesian. Now, the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from Corollary \ref{cor:usefulcriterion} that $A_{(1,n)}$ is \good{}. Finally, since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a point, the fact that the previous square is Thomason homotopy cocartesian implies that $A_{(1,n)}$ has the homotopy type of a point. Similarly, for any $m \geq 0$, by considering the cocartesian square $... ... @@ -763,7 +763,7 @@ For any n \geq 0, consider the following cocartesian square A_{(m,1)} \ar[r] & A_{(m,n)}, \end{tikzcd}$ where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This $2$-functor is once again a folk cofibration, but it is \emph{not} in general a co-universal Thomason weak equivalence (it is if we make the hypothesis that $m\neq 0$, but we did not). However, since we made the hypothesis that $n\neq 0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta \to \Delta_n$ is a co-universal Thomason weak equivalence. Hence, the previous square is Thomason homotopy cocartesian and $A_{(m,n)}$ has the homotopy type of a point. Since $A_{(m,1)}$, $\Delta_1$ and $\Delta_n$ are \good{}, this shows that for $m \geq 0$ and $n >0$, $A_{(m,n)}$ is \good{}. where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This $2$-functor is once again a folk cofibration, but it is \emph{not} in general a co-universal Thomason equivalence (it is if we make the hypothesis that $m\neq 0$, but we did not). However, since we made the hypothesis that $n\neq 0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta \to \Delta_n$ is a co-universal Thomason equivalence. Hence, the previous square is Thomason homotopy cocartesian and $A_{(m,n)}$ has the homotopy type of a point. Since $A_{(m,1)}$, $\Delta_1$ and $\Delta_n$ are \good{}, this shows that for $m \geq 0$ and $n >0$, $A_{(m,n)}$ is \good{}. Similarly, if $m >0$ and $n\geq 0$, then $A_{(m,n)}$ has the homotopy type of a point and is \good{}. \end{paragr} ... ...
 ... ... @@ -47,7 +47,7 @@ Recall that for any $\oo$\nbd-category $C$, we write $p_C : C \to \sD_0$ the can %% We can now prove the main result of this section. \begin{proposition}\label{prop:contractibleisgood} Every contractible $\oo$\nbd-category $C$ is \good{} and we have Every oplax contractible $\oo$\nbd-category $C$ is \good{} and we have $\sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}$ ... ... @@ -61,17 +61,17 @@ Consider the commutative square \sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH(\sD_0). \end{tikzcd} \] It follows respectively from Proposition \ref{prop:oplaxhmptyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left morphisms of the above square are isomorphisms. Then, a simple explicit computation shows that $\sD_0$ is \good{} and that $\sH^{\pol}(\sD_0)\simeq \sH^{\sing}(\sD_0)\simeq \mathbb{Z}$. By a 2-out-of-3 property, we deduce that $\pi_C : \sH^{\sing}(C)\to \sH^{\pol}(C)$ is an isomorphism. It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left morphisms of the above square are isomorphisms. Then, a simple explicit computation shows that $\sD_0$ is \good{} and that $\sH^{\pol}(\sD_0)\simeq \sH^{\sing}(\sD_0)\simeq \mathbb{Z}$. By a 2-out-of-3 property, we deduce that $\pi_C : \sH^{\sing}(C)\to \sH^{\pol}(C)$ is an isomorphism. \end{proof} \begin{remark} Definition \ref{def:contractible} admits an expected lax'' variation and Proposition \ref{prop:contractibleisgood} is still true for lax contractible $\oo$\nbd-categories. Definition \ref{def:contractible} admits an obvious lax'' variation and Proposition \ref{prop:contractibleisgood} is also true for lax contractible $\oo$\nbd-categories. \end{remark} We end this section with an important result on slices $\oo$\nbd-category (Paragraph \ref{paragr:slices}). \begin{proposition}\label{prop:slicecontractible} Let $A$ be an $\oo$\nbd-category and $a_0$ an object of $A$. The $\oo$\nbd-category $A/a_0$ is contractible. Let $A$ be an $\oo$\nbd-category and $a_0$ an object of $A$. The $\oo$\nbd-category $A/a_0$ is oplax contractible. \end{proposition} \begin{proof} \todo{À écrire.} This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}. \end{proof} \section{Homology of globes and spheres} \begin{paragr} ... ...
 ... ... @@ -1151,10 +1151,10 @@ As an immediate consequence of the previous lemma, the functor $\lambda_{\leq n} \ho(\oo\Cat^{\folk}) \ar[r,"\LL \lambda"] & \ho(\Ch) \end{tikzcd} \] is \emph{not} commutative. If it were, then for every$n$\nbd-category$C$and every$k >n$, we would have$H_k^{pol}(\iota_n(C))=0$for every$k >n$, which is not even true for the case$n=1$in the following chapter. is \emph{not} commutative. If it were, then for every$n$\nbd-category$C$and every$k >n$, we would have$H_k^{\pol}(\iota_n(C))=0$for every$k >n$, which is not even true for the case$n=1$in the following chapter. \end{remark} A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the following corollary. \begin{corollary} \begin{corollary}\label{cor:polhmlgycofibrant} Let$n \geq 0$and$C$be an$\oo$\nbd-category. If$C$has a$k$\nbd-basis for every$ 0 \leq k \leq n-1$, then the canonical map of$\ho(\Ch)$$\alpha^{\pol}_C : \sH^{\pol}(C) \to \lambda(C) ... ... @@ -1168,7 +1168,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi \begin{proof} \todo{À écrire} \end{proof} \begin{paragr} \begin{paragr}\label{paragr:polhmlgylowdimension} Since every \oo\nbd-category trivially admits its set of 0\nbd-cells as a 0\nbd-base, it follows from the previous proposition that for every \oo\nbd-category C we have \[ \sH^{\pol}_0(C)\simeq H_0(\lambda(C)) ... ... @@ -1264,7 +1264,7 @@ Straighforward consequence of the fact that N_n = N_{\oo} \circ \iota_n and th and a thorough analysis of naturality shows that this isomorphism is nothing but the canonical morphism c_1N_{\oo}(C) \to \tau_{\leq 1}^{i}(C). \end{proof} We can now prove the important following proposition. \begin{proposition} \begin{proposition}\label{prop:singhmlgylowdimension} For every \oo\nbd-category C, the canonical map of \ho(\Ch) \[ \alpha^{\sing}: \sH^{\sing}(C) \to \lambda(C) ... ... @@ -1282,7 +1282,7 @@ We can now prove the important following proposition.$ induced by the co-unit of the adjunction$c_{\oo} \dashv N_{\oo}$. From \ref{prop:polhmlgytruncation} we have that $\tau^{i}_1\lambda c_{\oo}N_{\oo}(C) \simeq \lambda_{\leq 1} \tau_{\leq 1}^{i} c_{\oo} N_{\oo}(C)=\lambda_{\leq 1} c_1 N_{\oo}(C), \tau^{i}_{\leq 1}\lambda c_{\oo}N_{\oo}(C) \simeq \lambda_{\leq 1} \tau_{\leq 1}^{i} c_{\oo} N_{\oo}(C)=\lambda_{\leq 1} c_1 N_{\oo}(C),$ and from Lemma \ref{lemma:truncationcounit} we obtain $... ... @@ -1291,7 +1291,7 @@ We can now prove the important following proposition. This means exactly that the image by \overline{\tau^{i}_{\leq 1}} of \alpha^{\sing} is an isomorphism, which is what we wanted to prove. \end{proof} Finally, we obtain the result we were aiming for. \begin{proposition} \begin{proposition}\label{prop:comphmlgylowdimension} For every \oo\nbd-category C, the canonical comparison map \[ \pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C) ... ... @@ -1302,6 +1302,39 @@ Finally, we obtain the result we were aiming for.$ for$k \in \{0,1\}$. \end{proposition} \begin{proof} Let$C$be an$\oo$\nbd-category and consider the following commutative triangle of$\ho(\Ch)$$\begin{tikzcd}[column sep=tiny] \sH^{\sing}(C) \ar[rd,"\alpha^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha^{\pol}"] \\ &\lambda(C)& \end{tikzcd}.$ From Proposition \ref{prop:singhmlgylowdimension}, we know that$\alpha^{\sing}$induces isomorphisms$H_k^{\sing}(C) \simeq H_k(\lambda(C))$for$k \in \{0,1\}$and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that$\alpha^{\pol}$induces isomorphisms$H_k^{\pol}(C) \simeq H_k(\lambda(C))$for$k \in \{0,1\}$. The result follows then from an immediate 2-out-of-3 property. \end{proof} \begin{paragr} A natural question following the above proposition is: \begin{center} For which$k \geq 0$do we have$H_k^{\sing}(C) \simeq H_k^{\pol}(C)$for every$\oo$\nbd-category$C$? \end{center} We have already seen in \ref{paragr:bubble} that when$C = B^2\mathbb{N}$we have $H_{2p}^{\sing}(B^2\mathbb{N}) \not\simeq H^{\pol}_{2p}(B^2\mathbb{N})$ for every$p \geq 2$. Furthermore, with a similar argument to the one given in \ref{paragr:bubble}, we have that for every$k \geq 3$, the (nerve of the)$\oo$\nbd-category$B^k\mathbb{N}$is a$K(\mathbb{Z},k)$-space. In particular, we have $H_{2p+3}^{\sing}(B^{2p +1}\mathbb{N})\simeq \mathbb{Z}/{2\mathbb{Z}}$ for every$p \geq 1$(see \cite[Theorem 23.1]{eilenberg1954groups}). On the other hand, since$B^k\mathbb{N}$is a free$k$\nbd-category, we have$H_n^{\pol}(B^k\mathbb{N})=0$for all$n \geq k$. All in all, we have proved that for every$k \geq 4$, there exists at least one$\oo$\nbd-category$C$such that $H_k^{\sing}(C) \not\simeq H_k^{\pol}(C).$ However, it is still an open question to know whether for$k \in \{2,3\}$we have $H^{\sing}_k(C) \simeq H^{\pol}_k(C).$ The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of$k$is the analoguous of Lemma \ref{lemma:truncationcounit}. But contrary to the case$k=1$, it is not generally true that the canonical morphism$c_k N_{\oo}(C) \to \tau^{i}_{\leq k}(C)$is an isomorphism when$k \geq 2$. However, what we really need is that the image by$\lambda$of this morphism be a quasi-isomorphism. In the case$k=2$, it seems that this canonical morphism admits an oplax$2$\nbd-functor as an inverse up to oplax transformation which could be an hint towards the conjecture that$H^{\sing}_2(C) \simeq H^{\pol}_2(C)$for every$\oo$\nbd-category$C$. \end{paragr} %% Slightly less trivial is the following lemma. %% \begin{lemma} %% The following triangle of functors ... ... @@ -1315,7 +1348,3 @@ Finally, we obtain the result we were aiming for. %% \end{lemma} %%\section{Homology and Homotopy of$\oo$-categories in low dimension} %%% Local Variables: %%% mode: latex %%% TeX-master: "main" %%% End: No preview for this file type  ... ... @@ -16,7 +16,7 @@ \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. } ... ...  ... ... @@ -184,8 +184,16 @@ publisher = "Elsevier" publisher={ScienceDirect}, year={1995} } @article{eilenberg1954groups, title={On the groups {H}($\pi\$,n), {II}: Methods of computation}, author={Eilenberg, Samuel and MacLane, Saunders}, journal={Annals of Mathematics}, pages={49--139}, year={1954}, publisher={JSTOR} } @article{freyd1972categories, title={Categories of continuous functors, I}, title={Categories of continuous functors, {I}}, author={Freyd, Peter J and Kelly, G Max}, journal={Journal of pure and applied algebra}, volume={2}, ... ...
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