Commit 23f851d7 by Leonard Guetta

### edited a lot of typos

 \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 model structure where: 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 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 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, \item[-] the fibrations are the morphisms of chain complexes $f : X \to Y$ such that for every $n>0$, $f_n : X_n \to Y_n$ is an epimorphism. \end{itemize} ... ... @@ -527,7 +527,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins Let $\begin{tikzcd} f : y \ar[r,shift left]&z :g\ar[l,shift left] \end{tikzcd}$ be an adjunction and $h : x \to y$ an equivalence with quasi-inverse $k : y \to x$. Then $fh$ is left adjoint to $kg$. \end{lemma} We can now state and prove the promised result. \begin{proposition}\label{prop:hmlgyderived} \begin{theorem}\label{thm:hmlgyderived} Consider that $\oo\Cat$ is equipped with the Thomason equivalences. The abelianization functor $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and the left derived morphism $\LL \lambda^{\Th} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch) ... ... @@ -536,7 +536,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins \[ \sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).$ \end{proposition} \end{theorem} \begin{proof} Let $\nu$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}) and consider the following adjunctions $... ... @@ -589,7 +589,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins \ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow] \end{tikzcd}$ A thorough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition \ref{prop:hmlgyderived} enables us to give the following description of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism A thorough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Theorem \ref{thm:hmlgyderived} enables us to give the following description of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism $\lambda c_{\oo} N_{\oo} \Rightarrow \lambda.$ ... ...
 ... ... @@ -177,9 +177,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with $\gamma^{\Th} : n\Cat \to \Ho(n\Cat^{\Th})$ for the localization morphism. The reason is to avoid confusion with other weak equivalences on $n\Cat$ we will later introduce. for the localization morphism. The reason is to avoid confusion with other weak equivalences on $n\Cat$ which we will introduce later. \end{paragr} \begin{paragr} By definition, the nerve functor induces a morphism of localizers ${N_n : (n\Cat,\W_n^{\Th}) \to (\Psh{\Delta},\W_{\Delta})}$ and hence a morphism of op-prederivators ... ... @@ -822,7 +820,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences \end{align*} \end{paragr} \begin{proposition}\label{prop:folkthmA}(Folk Theorem $A$) Let \begin{theorem}\label{thm:folkthmA}(Folk Theorem $A$) Let $\begin{tikzcd}[column sep=small] A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\ ... ... @@ -834,7 +832,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences u/c_0 : A/c_0 \to B/c_0$ is an equivalence of $\oo$-categories, then so is $u$. \end{proposition} \end{theorem} \begin{proof} Before anything else, let us note the following trivial but important fact: for any $\oo$\nbd{}functor $F : X \to Y$ and any $n$-cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$. \begin{enumerate}[label=(\roman*)] ... ... @@ -895,7 +893,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences \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} \begin{proposition}[Ara and Maltsiniotis' Theorem A] Let \begin{theorem}[Ara and Maltsiniotis' Theorem A] Let $\begin{tikzcd}[column sep=small] A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\ ... ... @@ -907,7 +905,7 @@ The nerve functor N_{\omega} : \omega\Cat \to \Psh{\Delta} sends equivalences u/c_0 : A/c_0 \to B/c_0$ is a Thomason equivalence, then so is $u$. \end{proposition} \end{theorem} %%% Local Variables: ... ...
 ... ... @@ -439,8 +439,8 @@ for the category of (strict) $\oo$\nbd{}categories. weak equivalences for $\oo$\nbd{}categories and compare them. The two most significant new results to be found in this chapter are probably Proposition \ref{prop:folkisthom}, which states that every equivalence of $\oo$\nbd{}categories is a Thomason equivalence, and Proposition \ref{prop:folkthmA}, which states that equivalences of $\oo$\nbd{}categories $\oo$\nbd{}categories is a Thomason equivalence, and Theorem \ref{thm:folkthmA}, which states that equivalences of $\oo$\nbd{}categories satisfy a property reminiscent of Quillen's Theorem $A$ \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. ... ... @@ -451,7 +451,7 @@ for the category of (strict) $\oo$\nbd{}categories. Up to Section \ref{section:polygraphichmlgy} included, all the results were known prior to this thesis (at least in the folklore), but starting from Section \ref{section:singhmlgyderived} all the results are original. Three fundamental results of this chapter are: Proposition \ref{prop:hmlgyderived}, fundamental results of this chapter are: Theorem \ref{thm:hmlgyderived}, which states that singular homology is obtained as a derived functor of an abelianization function, Proposition \ref{prop:criteriongoodcat}, which gives an abstract criterion to detect \good{} $\oo$\nbd{}categories, and Proposition ... ...