### Edited a lot of typos. This should almost be it.

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 ... ... @@ -18,8 +18,8 @@ In this section, we review some homotopical results on free The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and \emph{morphism of reflexive graphs} $f : G \to G'$ consists of maps $f_0 \colon G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they are the morphisms $f : G \to G'$ that are injective on objects and on arrows, ... ... @@ -264,9 +264,9 @@ From the previous proposition, we deduce the following very useful corollary. \] This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that the monomorphisms are the cofibrations of the standard Quillen model structure on simplicial sets and from Lemma \ref{lemma:hmtpycocartesianreedy}. result follows from Lemma \ref{lemma:hmtpycocartesianreedy} and the fact that the monomorphisms are the cofibrations of the standard Quillen model structure on simplicial sets. \end{proof} \begin{paragr} By working a little more, we obtain the more general result stated ... ... @@ -342,7 +342,8 @@ From the previous proposition, we deduce the following very useful corollary. \] where the arrow $E \to B$ is the canonical inclusion. Hence, by universal property, the dotted arrow exists and makes the whole diagram commute. A thorough verification easily shows that the morphism $G \to B$ is a thorough verification easily shows that, because $\beta$ is quasi-injective on arrows, the morphism $G \to B$ is a monomorphism of $\Rgrph$. By forming successive cocartesian squares and combining with the square ... ... @@ -482,10 +483,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \geq 0, we use the notations \begin{align*} X_{n,m} &:= X([n],[m]) \\ \partial_i^h &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\ \partial_j^v &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\ s_i^h &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\ s_j^v&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}. \partial_i^v &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\ \partial_j^h &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\ s_i^v &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\ s_j^h&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}. \end{align*} The maps\partial_i^h$and$s_i^h$will be referred to as the \emph{horizontal} face and degeneracy operators; and$\partial_i^v$and ... ... @@ -548,8 +549,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition definition,$\delta^*$induces a morphism of op-prederivators $\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta}). \Ho(\Psh{\Delta}),$ where$\Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}})$is the homotopy op-prederivator of$\Psh{\Delta\times \Delta}$equipped with diagonal weak equivalences. Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are the diagonal weak equivalences and whose fibrations are the ... ... @@ -1120,15 +1123,15 @@ of$2$\nbd{}categories. is \emph{not} \good{} (see \ref{paragr:bubble}). \end{paragr} \begin{paragr} For$n\geq 0$, we write$\Delta_n$for the linear order${0 \leq \cdots \leq n}$seen as a small category. Let$i : \Delta_1 \to \Delta_n$be the unique For$n\geq 0$, we write$\Delta_n$for the linear order$\{0 \leq \cdots \leq n\}$seen as a small category. Let$i : \Delta_1 \to \Delta_n$be the unique functor such that $i(0)=0 \text{ and } i(1)=n.$ \end{paragr} \begin{lemma}\label{lemma:istrngdefrtract} For$n\neq0$, the functor$i : \Delta_1 \to \Delta_n$is a strong deformation For$n\neq0$, the functor$i : \Delta_1 \to \Delta_n$is a strong oplax deformation retract (\ref{paragr:defrtract}). \end{lemma} \begin{proof} ... ... @@ -1156,7 +1159,7 @@ of$2$\nbd{}categories. \] where$\tau : \Delta_1 \to A_{(1,1)}$is the$2$\nbd{}functor that sends the unique non-trivial$1$\nbd{}cell of$\Delta_1$to the target of the generating$2$\nbd{}cell of$A_{(1,1)}$. It is not hard to check that$\tau$is strong$2$\nbd{}cell of$A_{(1,1)}$. It is not hard to check that$\tau$is strong oplax deformation retract and thus, a co-universal Thomason equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism$\Delta_n \to A_{(1,n)}$is also a (co-universal) Thomason equivalence and the square is ... ... @@ -1531,11 +1534,15 @@ Let us now get into more sophisticated examples. \item$G(A')=A$, \item$G(\gamma)=\alpha\comp_1\beta$. \end{itemize} Notice that we have$F\circ G = \mathrm{id}_{P'}$, which means that$P'$is a retract of$P$. In particular,$\sH^{\sing}(P)$is a retract of$\sH^{\sing}(P')$and since$P'$has the homotopy type of a$K(\mathbb{Z},2)$(see \ref{paragr:bubble}), this proves that$P$has non-trivial singular homology groups in all even dimension. But since it is a Notice that we have$F\circ G = \mathrm{id}_{P'}$and that we have an oplax transformation $h \colon \mathrm{id}_{P} \Rightarrow G \circ F$ defined as \begin{itemize}[label=-] \item$h_A:=1_A$, \item$h_f:=\alpha$. \end{itemize} Hence,$F$is a Thomason equivalence (Proposition \ref{prop:oplaxhmtpyisthom}) and$P$has the homotopy type of a$K(\mathbb{Z},2)$(see \ref{paragr:bubble}). In particular, it has non-trivial singular homology groups in all even dimension; but since it is a free$2$\nbd{}category, all its polygraphic homology groups are trivial strictly above dimension$2$, which means that$P$is \emph{not} \good{}. \end{paragr} ... ... @@ -1703,7 +1710,7 @@ Now let$\sS_2$be labelled as$\sS_2$. Notice that we have a cocartesian square \begin{equation}\label{square:bouquet} \begin{tikzcd} \sD_1 \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta \rangle"] & \sD_2 \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta \rangle"] & P' \ar[d] \\ P'' \ar[r] & P, \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd} ... ... @@ -1712,19 +1719,19 @@ Now let$\sS_2$be labelled as show that the square induced by the nerve $\begin{tikzcd} N_{\oo}(\sD_1) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta N_{\oo}(\sD_2) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle \beta \rangle"] & N_{\oo}(P') \ar[d] \\ N_{\oo}(P'') \ar[r] & N_{\oo}(P) \end{tikzcd}$ is also cocartesian. Since$\langle \beta \rangle : \sD_1 \to P'$and$\langle \beta \rangle : \sD_1 \to P''$are monomorphisms and$N_{\oo}$preserves is also cocartesian. Since$\langle \beta \rangle : \sD_2 \to P'$and$\langle \beta \rangle : \sD_2 \to P''$are monomorphisms and$N_{\oo}$preserves monomorphisms, it follows from Lemma \ref{lemma:hmtpycocartesianreedy} that square \eqref{square:bouquet} is Thomason homotopy cocartesian and in particular that$P$has the homotopy type of a bouquet of two$2$\nbd{}spheres. Since$\sD_1$,$P'$and$P''$are free and \good{} and since$\langle \beta \rangle : \sD_1 \to P'$and$\langle \beta \rangle : \sD_1 \to $2$\nbd{}spheres. Since $\sD_2$, $P'$ and $P''$ are free and \good{} and since $\langle \beta \rangle : \sD_2 \to P'$ and $\langle \beta \rangle : \sD_2 \to P''$ are folk cofibrations, this also proves that $P$ is \good{} (see \ref{paragr:criterion2cat}). \end{paragr} \begin{paragr} ... ... @@ -1905,7 +1912,7 @@ Now let $\sS_2$ be labelled as \item $(1_f,\cdots,1_f)$, \item $(1_g,\cdots,1_g)$, \end{itemize} whose generating arrows from $B$ to $C$ are $k$\nbd{}tuple of one of the whose arrows from $B$ to $C$ are $k$\nbd{}tuple of one of the following form \begin{itemize}[label=-] \item $(1_h,\cdots,1_h,\gamma,1_i,\cdots,1_i)$, ... ... @@ -1950,9 +1957,9 @@ homotopy type of the torus. f \cdots fg\cdots g \] where $f$ is repeated $n$ times and $g$ is repeated $m$ times. Recall that the category $B^1(\mathbb{N}\times\mathbb{N})$ is nothing but the Recall that we write $B^1(\mathbb{N}\times\mathbb{N})$ for the monoid $\mathbb{N}\times\mathbb{N}$ considered as a category with only one object, and let $F : P \to B^1(\mathbb{N}\times\mathbb{N})$ be the unique object, and let $F : P \to B^1(\mathbb{N}\times\mathbb{N})$ be the unique $2$\nbd{}functor such that: \begin{itemize}[label=-] \item $F(f)=(1,0)$ and $F(g)=(0,1)$, ... ... @@ -2073,7 +2080,7 @@ homotopy type of the torus. \begin{itemize}[label=-] \item $x$ is not a unit, \item $\src_0(x)=\trgt_0(x)$, \item $\trgt(x)=\src(x)=1_{\src_0(x)}$. \item $\trgt_1(x)=\src_1(x)=1_{\src_0(x)}$. \end{itemize} \end{definition} \begin{paragr} ... ...
 ... ... @@ -2,7 +2,8 @@ consequences} \chaptermark{Contractible $\omega$-categories and consequences} \section{Contractible \texorpdfstring{$\oo$}{ω}-categories} Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ for the canonical morphism to the terminal object of $\sD_0$. Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ for the canonical morphism to the terminal object $\sD_0$ of $\oo\Cat$. \begin{definition}\label{def:contractible} An $\oo$\nbd{}category $C$ is \emph{oplax contractible} when the canonical morphism $p_C : C \to \sD_0$ is an oplax homotopy equivalence (Definition \ref{def:oplaxhmtpyequiv}). ... ... @@ -60,7 +61,7 @@ Consider the commutative square $\begin{tikzcd} \sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\ \sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0). \sH^{\pol}(\sD_0) \ar[r,"\pi_{\sD_0}"] & \sH^{\sing}(\sD_0). \end{tikzcd}$ It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and ... ... @@ -96,9 +97,9 @@ We end this section with an important result on slice $\oo$\nbd{}categories (Par Now for $0 \leq k  ... ... @@ -1222,7 +1222,7 @@ As a consequence of this lemma, we have the analogous of Proposition \ref{prop:t \] where$\eta$is the unit map of the adjunction$\tau^{i}_{\leq n} \dashv \iota_n$. It follows from Lemma \ref{lemma:unitajdcomp} that $H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to \iota_n\tau^{i}_{\leq n}(K')) H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to H_k(\iota_n\tau^{i}_{\leq n}(K'))$ is an isomorphism for every$k \leq n$. Since obviously$H_k(\iota_n\tau^{i}_{\leq n}(f))$is also an isomorphism for$k > n$, this proves the result. \end{proof} ... ... @@ -1370,7 +1370,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi for every$0 \leq k \leq n$. \end{corollary} \begin{proof} From Lemma \ref{lemma:abelianizationtruncation} and Proposition \ref{prop:polhmlgytruncation}, we deduce that the morphism$\overline{\tau^{i}_{\leq n}}(\alpha^{\pol})$of$\ho(\Ch^{\leq n})$can be identified with the canonical morphism From Lemma \ref{lemma:abelianizationtruncation} and Proposition \ref{prop:polhmlgytruncation}, we deduce that the morphism$\overline{\tau^{i}_{\leq n}}(\alpha_C^{\pol})$of$\ho(\Ch^{\leq n})$can be identified with the canonical morphism $\LL \lambda_{\leq n}(\tau^{i}_{\leq n}(C)) \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C)).$ ... ... @@ -1421,7 +1421,7 @@ and for$n \in \mathbb{N}\cup \{\oo\}$we write$c_n : \Psh{\Delta} \to n\Cat$f Straightforward consequence of the fact that$N_n = N_{\oo} \circ \iota_n$and the fact that the composition of left adjoints is the left adjoint of the composition. \end{proof} \begin{paragr} In particular, it follows from the previous lemma that the co-unit of the adjunction$c_{\oo} \dashv N_{\oo}$induces, for every$\oo$\nbd{}category$C$and every$n \geq 0$, a canonical morphism of$n\Cat$In particular, it follows from the previous lemma that the co-unit of the adjunction$c_{\oo} \dashv N_{\oo}$induces for every$\oo$\nbd{}category$C$and every$n \geq 0$, a canonical morphism of$n\Cat$$c_nN_{\oo}(C) \simeq \tau^{i}_{\leq n}c_{\oo}N_{\oo}(C) \to \tau^{i}_{\leq n}(C),$ ... ... @@ -1449,7 +1449,7 @@ Straightforward consequence of the fact that$N_n = N_{\oo} \circ \iota_nand t &\simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))). \end{align*} Using the description of\Or_0$,$\Or_1$and$\Or_2$from \ref{paragr:orientals}, we deduce that a morphism$F : i^*(N_{\oo}(C)) \to i^*(N_1(D))$of$\Psh{\Delta_{\leq 2}}$consists of a function$F_0 : C_0 \to D_0$and a function$F_1 : C_1 \to D_1$such that \begin{itemize}[label=-] \begin{enumerate}[label=(\alph*)] \item for every$x \in C_0$, we have$F_1(1_x)=1_{F_0(x)}$, \item for every$x \in C_1$, we have $\src(F_1(x))=F_0(\src(x))) \text{ and }\trgt(F_1(x))=F_0(\trgt(x))),$ ... ... @@ -1462,8 +1462,11 @@ Straightforward consequence of the fact that$N_n = N_{\oo} \circ \iota_n$and t \end{tikzcd} \] in$C$, we have$F_1(g)\comp_0 F_1(f)=F_1(h)$. \end{itemize} In particular, it follows that$F_1$is compatible with composition of$1$\nbd{}cells in an obvious sense and that for every$2$\nbd{}cell$\alpha : f \Rightarrow g$of$C$, we have$F_1(f)=F_1(g)$. This means exactly that we have a natural isomorphism \end{enumerate} In particular, it follows that$F_1$is compatible with composition of$1$\nbd{}cells in an obvious sense and that for every$2$\nbd{}cell$\alpha : f \Rightarrow g$of$C$, we have$F_1(f)=F_1(g)$. And conversely, this last condition implies condition (c) above. This means exactly that we have a natural isomorphism $\Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))) \simeq \Hom_{\Cat}(\tau_{\leq 1}^{i}(C),D).$ ... ... @@ -1481,7 +1484,7 @@ We can now prove the important following 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) \alpha_C^{\sing}: \sH^{\sing}(C) \to \lambda(C)$ induces isomorphisms $... ... @@ -1490,7 +1493,7 @@ We can now prove the important following proposition. for k \in \{0,1\}. \end{proposition} \begin{proof} Let C be an \oo\nbd{}category. Recall from \ref{paragr:univmor} that the canonical morphism \alpha^{\sing} : \sH^{\sing}(C) \to \lambda(C) is nothing but the image by the localization functor \Ch \to \ho(\Ch) of the morphism Let C be an \oo\nbd{}category. Recall from \ref{paragr:univmor} that the canonical morphism \alpha_C^{\sing} : \sH^{\sing}(C) \to \lambda(C) is nothing but the image by the localization functor \Ch \to \ho(\Ch) of the morphism \[ \lambda c_{\oo}N_{\oo}(C) \to \lambda(C)$ ... ... @@ -1502,7 +1505,7 @@ We can now prove the important following proposition. $\tau_{\leq 1 }^{i} \lambda c_{\oo} N_{\oo}(C) \simeq \lambda_{\leq 1} \tau^{i}_{\leq 1}(C) \simeq \tau^{i}_{\leq 1}\lambda(C).$ 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. This means exactly that the image by$\overline{\tau^{i}_{\leq 1}}$of$\alpha_C^{\sing}$is an isomorphism, which is what we wanted to prove. \end{proof} Finally, we obtain the result we were aiming for. \begin{proposition}\label{prop:comphmlgylowdimension} ... ... @@ -1520,11 +1523,11 @@ Finally, we obtain the result we were aiming for. 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}"] \\ \sH^{\sing}(C) \ar[rd,"\alpha_C^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha_C^{\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. From Proposition \ref{prop:singhmlgylowdimension}, we know that$\alpha_C^{\sing}$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$\alpha_C^{\pol}$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}\label{paragr:conjectureH2} A natural question following the above proposition is: ... ... No preview for this file type  ... ... @@ -9,7 +9,7 @@ publisher={Narnia} } @article{ara2014vers, title={Vers une structure de cat{\'e}gorie de mod{\e}les {\a} la {T}homason sur la cat{\'e}gorie des n\nbd{}cat{\'e}gories strictes}, title={Vers une structure de cat{\'e}gorie de mod{\e}les {\a} la {T}homason sur la cat{\'e}gorie des$n$\nbd{}cat{\'e}gories strictes}, author={Ara, Dimitri and Maltsiniotis, Georges}, journal={Advances in Mathematics}, volume={259}, ... ... @@ -40,7 +40,7 @@ year={2020} } @article{ara2020theoreme, title={Un th{\'e}or{\e}me {A} de {Q}uillen pour les$\infty$-cat{\'e}gories strictes {II} : la preuve$\infty$-cat{\'e}gorique}, title={Un th{\'e}or{\e}me {A} de {Q}uillen pour les$\infty$\nbd{}cat{\'e}gories strictes {II} : la preuve$\infty\$-cat{\'e}gorique}, author={Ara, Dimitri and Maltsiniotis, Georges}, volume={4}, number={1}, ... ...
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