### Encore des typos et du layout

parent 5df9566f
 ... ... @@ -25,7 +25,7 @@ In this section, we review some homotopical results on free are the morphisms $f : G \to G'$ that are injective on objects and on arrows, i.e. such that $f_0 : G_0 \to G_0'$ and $f_1 : G_1 \to G'_1$ are injective. There is a underlying reflexive graph'' functor There is an underlying reflexive graph'' functor $U : \Cat \to \Rgrph,$ ... ... @@ -239,7 +239,7 @@ From the previous proposition, we deduce the following very useful corollary. L(C) \ar[r,"L(\gamma)"]& L(D) \end{tikzcd} \] is a Thomason homotopy cocartesian. is Thomason homotopy cocartesian. \end{corollary} \begin{proof} Since the nerve $N$ induces an equivalence of op-prederivators ... ... @@ -272,11 +272,11 @@ From the previous proposition, we deduce the following very useful corollary. By working a little more, we obtain the more general result stated in the proposition below. Let us say that a morphism of reflexive graphs $\alpha : A \to B$ is \emph{quasi-injective on arrows} when for all $f$ and $g$ arrows of $A$, if for all arrows $f$ and $g$ of $A$, if $\alpha(f)=\alpha(g),$ then either $f=g$ or $f$ and $g$ are both units. In other words, $\alpha$ then either $f=g$, or $f$ and $g$ are both units. In other words, $\alpha$ never sends a non-unit arrow to a unit arrow and $\alpha$ never identifies two non-unit arrows. It follows that if $\alpha$ is quasi-injective on arrows and injective on objects, then it is also injective on arrows and hence, a ... ... @@ -304,7 +304,7 @@ From the previous proposition, we deduce the following very useful corollary. L(C) \ar[r,"L(\gamma)"] &L(D) \end{tikzcd} \] is Thomason homotopy cocartesian square. is Thomason homotopy cocartesian. \end{proposition} \begin{proof} The case where $\alpha$ or $\beta$ is both injective on objects and ... ... @@ -415,7 +415,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{cor:hmtpysquaregraph}. \end{example} \begin{remark} Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration% , since a Thomason homotopy Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration % , since a Thomason homotopy % cocartesian square in $\Cat$ is also so in $\oo\Cat$ and since every free category is obtained by recursively adding generators starting from a set of objects (seen as a $0$-category), the previous example ... ... @@ -456,7 +456,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd} \] Then, this above square is Thomason homotopy cocartesian. Indeed, it Then, this square is Thomason homotopy cocartesian. Indeed, it obviously is the image of a cocartesian 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 $\Rgrph$. Hence, we can ... ... @@ -744,7 +744,7 @@ In practice, we will use the following corollary. \ref{prop:bisimplicialcocontinuous}. \end{proof} \section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve} We shall now describe a nerve'' for $2$-categories with values in bisimplicial We shall now describe a nerve'' for $2$\nbd{}categories with values in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the nerve defined in \ref{paragr:nerve}. \begin{notation} ... ... @@ -757,23 +757,23 @@ equivalent to the nerve defined in \ref{paragr:nerve}. \] is the set of $k$-simplices of the nerve of $C$. \item[-] Similarly, we write $N : 2\Cat \to \Psh{\Delta}$ instead of $N_2$ for the nerve of $2$-categories. This makes sense since the nerve for categories is the restriction of the nerve for $2$-categories. \item[-] For $2$-categories, we refer to the $\comp_0$-composition of $2$-cells as the \emph{horizontal composition} and the $\comp_1$-composition of $2$-cells as the \emph{vertical composition}. \item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by the nerve of $2$\nbd{}categories. This makes sense since the nerve for categories is the restriction of the nerve for $2$\nbd{}categories. \item[-] For $2$\nbd{}categories, we refer to the $\comp_0$-composition of $2$\nbd{}cells as the \emph{horizontal composition} and the $\comp_1$-composition of $2$\nbd{}cells as the \emph{vertical composition}. \item[-] For a $2$\nbd{}category $C$ and $x$ and $y$ objects of $C$, we denote by $C(x,y)$ the category whose objects are the $1$-cells of $C$ with $x$ as source and $y$ as target, and whose arrows are the $2$-cells of $C$ with $x$ as $y$ as target, and whose arrows are the $2$\nbd{}cells of $C$ with $x$ as $0$-source and $y$ as $0$-target. Composition is induced by vertical composition in $C$. \end{itemize} \end{notation} \begin{paragr} Every $2$-category $C$ defines a simplicial object in $\Cat$, Every $2$\nbd{}category $C$ defines a simplicial object in $\Cat$, $S(C): \Delta^{\op} \to \Cat,$ where, for each $n \geq 0$, $S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1) ... ... @@ -795,7 +795,7 @@ equivalent to the nerve defined in \ref{paragr:nerve}. category. \end{remark} \begin{definition} The \emph{bisimplicial nerve} of a 2-category C is the bisimplicial set The \emph{bisimplicial nerve} of a 2\nbd{}category C is the bisimplicial set \binerve(C) defined as \[ \binerve(C)_{n,m}:=N(S_n(C))_m, ... ... @@ -804,7 +804,7 @@ equivalent to the nerve defined in \ref{paragr:nerve}. \end{definition} \begin{paragr}\label{paragr:formulabisimplicialnerve} In other words, the bisimplicial nerve of C is obtained by un-currying'' the functor NS(C) : \Delta^{op} \to \Psh{\Delta}. the functor NS(C) : \Delta^{\op} \to \Psh{\Delta}. Since the nerve N commutes with products and sums, we obtain the formula \begin{equation}\label{fomulabinerve} ... ... @@ -847,7 +847,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio \[ \alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)$ of vertically composable $2$-cells of $C$, i.e.\ such that for of vertically composable $2$\nbd{}cells of $C$, i.e.\ such that for every $1 \leq i \leq k-1$, we have $\src(\alpha_i)=\trgt(\alpha_{i+1}). ... ... @@ -879,7 +879,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio degeneracy operators are induced by the units for the vertical composition. \end{paragr} \begin{lemma}\label{lemma:binervehorizontal} Let C be a 2-category. For every n \geq 0, we have Let C be a 2\nbd{}category. For every n \geq 0, we have \[ N(V_m(C))_n=(\binerve(C))_{n,m}.$ ... ... @@ -897,7 +897,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio theory of bisimplicial sets. \end{paragr} \begin{lemma}\label{lemma:binervthom} A $2$-functor $F : C \to D$ is a Thomason equivalence if and only if A $2$\nbd{}functor $F : C \to D$ is a Thomason equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets. \end{lemma} \begin{proof} ... ... @@ -910,9 +910,9 @@ an equivalent definition of the bisimplicial nerve which uses the other directio \cite[Théorème 3.13]{ara2020comparaison}. \end{proof} From this lemma, we deduce two useful criteria to detect Thomason equivalences of $2$-categories. of $2$\nbd{}categories. \begin{corollary}\label{cor:criterionThomeqI} Let $F : C \to D$ be a $2$-functor. If Let $F : C \to D$ be a $2$\nbd{}functor. If \begin{enumerate}[label=\alph*)] \item $F_0 : C_0 \to D_0$ is a bijection, \end{enumerate} ... ... @@ -924,10 +924,10 @@ of $2$-categories. \] induced by $F$ is a Thomason equivalence of $1$-categories, \end{enumerate} then $F$ is a Thomason equivalence of $2$-categories. then $F$ is a Thomason equivalence of $2$\nbd{}categories. \end{corollary} \begin{proof} By definition, for every $2$-category $C$ and every $m \geq 0$, we have By definition, for every $2$\nbd{}category $C$ and every $m \geq 0$, we have $(\binerve(C))_{\bullet,m} = NS(C).$ ... ... @@ -936,9 +936,9 @@ of $2$-categories. products. \end{proof} \begin{corollary}\label{cor:criterionThomeqII} Let $F : C \to D$ be a $2$-functor. If for every $k \geq 0$, Let $F : C \to D$ be a $2$\nbd{}functor. If for every $k \geq 0$, $V_k(F) : V_k(C) \to V_k(D)$ is a Thomason equivalence of $1$-categories, then $F$ is a Thomason equivalence of $2$-categories. then $F$ is a Thomason equivalence of $2$\nbd{}categories. \end{corollary} \begin{proof} From Lemma \ref{lemma:binervehorizontal}, we now that for every $m \geq 0$, ... ... @@ -1071,8 +1071,8 @@ of $2$-categories. cocartesian, then $D$ is \good{}. \end{paragr} \begin{paragr} Let $n,m \geq 0$. We denote by $A_{(m,n)}$ the free $2$-category with only one generating $2$-cell whose source is a chain of length $m$ and whose target is a Let $n,m \geq 0$. We denote by $A_{(m,n)}$ the free $2$\nbd{}category with only one generating $2$\nbd{}cell whose source is a chain of length $m$ and whose target is a chain of length $n$: $\underbrace{\overbrace{\begin{tikzcd}[column sep=small, ampersand ... ... @@ -1092,7 +1092,7 @@ of 2-categories. for } 0\leq i \leq m-1 \\ g_{j+1} : B_j \to B_{j+1} &\text{ for } 0 \leq j \leq n-1 %\\g_1 : A_0 \to B_1 & \\g_{n} : B_{n-1} \to A_m & \end{cases} \item generating 2-cell:  \alpha : f_{m}\circ \cdots \circ f_1 \Rightarrow \item generating 2\nbd{}cell:  \alpha : f_{m}\circ \cdots \circ f_1 \Rightarrow g_n \circ \cdots \circ g_1. \end{itemize} Notice that A_{(1,1)} is nothing but \sD_2. We are going to prove that if ... ... @@ -1113,10 +1113,10 @@ of 2-categories. \ar[from=A,to=1-1,Rightarrow,"\alpha"] \end{tikzcd}$ and has many non trivial $2$-cells, such as $f\comp_0 \alpha \comp_0 f$. and has many non trivial $2$\nbd{}cells, such as $f\comp_0 \alpha \comp_0 f$. Note that when $m=0$ \emph{and} $n=0$, then the $2$-category $A_{(0,0)}$ is nothing but the $2$-category $B^2\mathbb{N}$ and we have already seen that it Note that when $m=0$ \emph{and} $n=0$, then the $2$\nbd{}category $A_{(0,0)}$ is nothing but the $2$\nbd{}category $B^2\mathbb{N}$ and we have already seen that it is \emph{not} \good{} (see \ref{paragr:bubble}). \end{paragr} \begin{paragr} ... ... @@ -1154,9 +1154,9 @@ of $2$-categories. end,"\ulcorner"] \end{tikzcd} \] where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique 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 $2$-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 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 ... ... @@ -1176,7 +1176,7 @@ of $2$-categories. end,"\ulcorner"] \end{tikzcd} \] where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$\nbd{}functor that sends the unique non trivial $1$\nbd{}cell of $\Delta_1$ to the source of the generating $2$\nbd{}cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has the homotopy type of a point. ... ... @@ -1190,9 +1190,9 @@ of $2$-categories. end,"\ulcorner"] \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 where $\tau$ is the $2$\nbd{}functor that sends the unique non-trivial $1$-cell of $\Delta_1$ to the target of the generating $2$\nbd{}cell of $A_{(m,1)}$. This $2$\nbd{}functor is once again a folk cofibration, but it is \emph{not} in general a co-universal Thomason equivalence (it would be if we had made 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_1 \to ... ... @@ -1207,7 +1207,7 @@ of$2$-categories. Combined with the result of Paragraph \ref{paragr:bubble}, we have proved the following proposition. \begin{proposition}\label{prop:classificationAmn} Let$m,n \geq 0$and consider the$2$-category$A_{(m,n)}$. If$m\neq 0$or Let$m,n \geq 0$and consider the$2$\nbd{}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$K(\mathbb{Z},2)$. ... ... @@ -1546,7 +1546,7 @@ Let us now get into more sophisticated examples. \begin{center} \begin{tabular}{| l || c | c |} \hline$2$-category & \good{}? & homotopy type \\ \hline \hline$2$\nbd{}category & \good{}? & homotopy type \\ \hline \hline {$\begin{tikzcd} \bullet \ar[r,bend ... ...
 ... ... @@ -182,11 +182,11 @@ homotopy theory, polygraphs. \pagestyle{fancy} \fancyhf{} \fancyfoot[C]{\thepage} \fancyhead[RO]{INTRODUCTION} \fancyhead[RO,LE]{INTRODUCTION} \include{introduction} \fancyhf{} \fancyfoot[C]{\thepage} \fancyhead[RO]{INTRODUCTION (FRANÇAIS)} \fancyhead[RO,LE]{INTRODUCTION (FRANÇAIS)} \include{introduction_fr} \fancyhf{} \fancyfoot[C]{\thepage} ... ...
 ... ... @@ -57,7 +57,7 @@ year={2020} year={2020} } @article{ara2019quillen, title={A {Q}uillen Theorem {B} for strict $\infty$\nbd{}categories}, title={A {Q}uillen {T}heorem {B} for strict $\infty$\nbd{}categories}, author={Ara, Dimitri}, journal={Journal of the London Mathematical Society}, volume={100}, ... ... @@ -67,7 +67,7 @@ year={2020} publisher={Wiley Online Library} } @article{batanin1998monoidal, title={Monoidal globular categories as a natural environment for the theory of weak n\nbd{}categories}, title={Monoidal globular categories as a natural environment for the theory of weak $n$\nbd{}categories}, author={Batanin, Michael A.}, journal={Advances in Mathematics}, volume={136}, ... ... @@ -137,7 +137,7 @@ publisher = "Elsevier" publisher={Citeseer} } @article{cisinski2003images, title={Images directes cohomologiques dans les cat{\'e}gories de modeles}, title={Images directes cohomologiques dans les cat{\'e}gories de mod{\`e}les}, author={Cisinski, Denis-Charles}, journal={Annales Math{\'e}matiques Blaise Pascal}, volume={10}, ... ... @@ -170,7 +170,7 @@ publisher = "Elsevier" year={1995} } @article{duskin2002simplicial, title={Simplicial matrices and the nerves of weak n\nbd{}categories. {I}. Nerves of bicategories}, title={Simplicial matrices and the nerves of weak $n$\nbd{}categories. {I}. Nerves of bicategories}, author={Duskin, John W.}, journal={Theory and Applications of Categories}, volume={9}, ... ... @@ -197,7 +197,7 @@ publisher = "Elsevier" year={1995} } @article{eilenberg1954groups, title={On the groups {H}($\pi$,n), {II}: Methods of computation}, title={On the groups ${H}(\pi,n)$, {II}: Methods of computation}, author={Eilenberg, Samuel and Mac Lane, Saunders}, journal={Annals of Mathematics}, pages={49--139}, ... ... @@ -221,7 +221,7 @@ publisher = "Elsevier" publisher={Springer} } @article{gagna2018strict, title={Strict n-categories and augmented directed complexes model homotopy types}, title={Strict $n$\nbd{}categories and augmented directed complexes model homotopy types}, author={Gagna, Andrea}, journal={Advances in Mathematics}, volume={331}, ... ... @@ -480,7 +480,7 @@ note={In preparation} publisher={Springer} } @article{schreiber2013differential, title={Differential cohomology in a cohesive infinity-topos}, title={Differential cohomology in a cohesive $\infty$\nbd{}topos}, author={Schreiber, Urs}, journal={arXiv preprint arXiv:1310.7930}, year={2013} ... ...
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