Commit 851f18d2 by Leonard Guetta

### edited typos

parent 6b0c8526
 ... ... @@ -98,7 +98,7 @@ In this section, we review some homotopical results on free if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all $n$-simplices of $X$ are degenerated for $n\geq 2$, then $f$ is a monomorphism. A proof of this assertion is contained in \cite[Paragraph 3.4]{gabriel1967calculus}. The key argument is the Eilenberg-Zilber Lemma 3.4]{gabriel1967calculus}. The key argument is the Eilenberg--Zilber Lemma (Proposition 3.1 of op. cit.). \end{proof} \begin{paragr} ... ... @@ -153,7 +153,7 @@ In this section, we review some homotopical results on free \] where $\eta$ is the unit of the adjunction $c \dashv N$. \end{paragr} \begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan} \begin{lemma}[Dwyer--Kan]\label{lemma:dwyerkan} For every $k\geq 1$, the canonical inclusion map $N^{k}(G) \to N^{k+1}(G) ... ... @@ -1617,7 +1617,7 @@ Now let \sS_2 be labelled as k=0. Similarly, for k>0, we have already seen that V_k(\sS_2) is the free category on the graph that has 2 objects and 2k+2 parallel arrows between these two objects and we leave as an easy exercice to the reader to check that the category V_k(P) is the exercise to the reader to check that the category V_k(P) is the free category on the graph that has one object and 2k+2 arrows, which are the k\nbd{}tuples of one of the following forms: \begin{itemize}[label=-] ... ...  \chapter{Homology and abelianization of \texorpdfstring{\oo}{ω}-categories} \section{Homology via the nerve} \begin{paragr} We denote by \Ch the category of non-negatively graded chain complexes of abelian groups. Recall that \Ch can be equipped with a cofibrantely generated model structure, known as the \emph{projective model structure on \Ch}, where: We denote by \Ch the category of non-negatively graded chain complexes of abelian groups. Recall that \Ch can be equipped with a cofibrantly generated model structure, known as the \emph{projective model structure on \Ch}, where: \begin{itemize} \item[-] the weak equivalences are the quasi-isomorphisms, i.e.\ morphisms of chain complexes that induce an isomorphism on homology groups, \item[-] the cofibrations are the morphisms of chain complexes f: X\to Y such that for every n\geq 0, f_n : X_n \to Y_n is a monomorphism with projective cokernel, ... ... @@ -1086,7 +1086,7 @@ We now turn to truncations of chain complexes. \end{itemize} \end{proposition} \begin{proof} This is a typical example of a transfer of a cofibrantely generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set J of generating trivial cofibrations of the projective model structure on \Ch such that for every j : A \to B in J and every cocartesian square This is a typical example of a transfer of a cofibrantly generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set J of generating trivial cofibrations of the projective model structure on \Ch such that for every j : A \to B in J and every cocartesian square \[ \begin{tikzcd} \tau^{i}_{\leq n}(A) \ar[r] \ar[d,"\tau^{i}_{\leq n}(j)"'] & X \ar[d,"g"] \\ ... ...  ... ... @@ -548,7 +548,7 @@ For later reference, we put here the following trivial but important lemma, whos \end{example} \begin{theorem}\label{thm:folkms} There exists a cofibrantely generated model structure on \omega\Cat such that the weak equivalences are the equivalences of \omega-categories, and the set \{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\} (see \ref{paragr:defglobe}) is a set of generating cofibrations. There exists a cofibrantly generated model structure on \omega\Cat such that the weak equivalences are the equivalences of \oo\nbd{}categories, and the set \{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\} (see \ref{paragr:defglobe}) is a set of generating cofibrations. \end{theorem} \begin{proof} This is the main result of \cite{lafont2010folk}. ... ... @@ -655,7 +655,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen %% is the identity on objects. This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of \oo\nbd{}categories. \end{paragr} \section{Slices of \texorpdfstring{\oo}{ω}-category and a folk Theorem A} \section{Slice \texorpdfstring{\oo}{ω}-categories and a folk Theorem A} \begin{paragr}\label{paragr:slices} Let A be an \oo\nbd{}category and a_0 an object of A. We define the slice \oo\nbd{}category A/a_0 as the following fibred product: \[ ... ... @@ -897,7 +897,6 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends the equivalen whose image by u/c_0 is equivalent for the relation \sim_{\oo} to the above (n+1)\nbd{}cell of B/c_0. In particular, the source and target of \alpha are respectively f and f'. Finally, we obtain that \alpha \sim_{\oo} \beta by applying the canonical \oo\nbd{}functor A/b_0 \to A, . \end{enumerate} \end{proof} %\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.} \begin{paragr} The name folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its \oo\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one. \end{paragr} ... ...  ... ... @@ -967,7 +967,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. (\M,\W,\Cof,\Fib) only depends on its underlying localizer, the existence of the classes \Cof and \Fib with the usual properties defining model structure ought to be thought as a \emph{property} of the localizer (\M,\W), which is sufficcient to define a homotopy theory''. For (\M,\W), which is sufficient to define a homotopy theory''. For example, Theorem \ref{thm:cisinskiI} should have been stated by saying that if a localizer (\M,\W) can be extended to a model category (\M,\W,\Cof,\Fib), then it is homotopy cocomplete. ... ... No preview for this file type  ... ... @@ -118,7 +118,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n$ whenever these equations make sense. A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have A \emph{morphism of $\oo$\nbd{}magmas} $f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have $f(1_x)=1_{f(x)},$ ... ... @@ -1623,7 +1623,7 @@ with $w_1$ and $w_2$ well formed words and $0\leq k \leq n$. Then $\src_k(w_1)=\ \section{Proof of Theorem \ref{thm:conduche}: part II} Recall that we have seen in Proposition \ref{prop:smallestcategoricalcongruence} that there exists a smallest categorical congruence on every$n$\nbd{}magma$X$with$n \geq 1$. However, the description of this congruence that we we used for with$n \geq 1$. However, the description of this congruence that we used for proving its existence is rather abstract. The main goal of this section is to give a more concrete description of the smallest categorical congruence in the case that$X= \E^+$for an$n$\nbd{}cellular extension$\E\$. This description ... ...
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