\chapter{Homology of contractible \texorpdfstring{$\oo$}{ω}-categories and its 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 $\sD_0$. \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}). %% there exists an object $x_0$ of $X$ and an oplax transformation %% $%% \begin{tikzcd} %% X \ar[r,"p_X"] \ar[rd,"1_X"',""{name=A,above}]& \sD_0 \ar[d,"\langle x_0 \rangle"]\\ %% &X. %% \ar[from=A,to=1-2,"\alpha",Rightarrow] %% \end{tikzcd} %%$ \end{definition} %% \begin{paragr} %% In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason equivalence. In particular, we have the following lemma. %% \end{paragr} %% \begin{lemma}\label{lemma:hmlgycontractible} %% Let $X$ be a contractible $\oo$\nbd{}category. The morphism of $\ho(\Ch)$ %% $%% \sH(X) \to \sH(\sD_0) %%$ %% induced by the canonical morphism $p_X : X \to \sD_0$ is an isomorphism. %% \end{lemma} %% \begin{paragr} %% In addition to the previous result, it is immediate to check that $\sH(\sD_0)$ is nothing but $\mathbb{Z}$ seen as an object of $\ho(\Ch)$ concentrated in degree $0$. %% \end{paragr} %% \begin{corollary} %% Let $u : X \to Y$ be a morphism of $\oo\Cat$. If $u$ is a homotopy equivalence (Paragraph \ref{paragr:hmtpyequiv}), then %% $%% \sH^{\pol}(\gamma_{\folk}(u)) %%$ %% is an isomorphism. %% \end{corollary} %% \todo{Expliquer p-e pourquoi le corollaire précédent est important.} %% \begin{corollary}\label{cor:hmlgypolcontractible} %% Let $X$ be an $\oo$-category. If $X$ is contractible, then the morphism %% $%% \sH^{\pol}(\gamma_{\folk}(p_X)) : \sH^{\pol}(X) \to \sH^{\pol}(\sD_0) %%$ %% is an isomorphism of $\ho(\Ch)$. %% \end{corollary} %% We can now prove the main result of this section. \begin{proposition}\label{prop:contractibleisgood} Every oplax contractible $\oo$\nbd{}category $C$ is \good{} and we have $\sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}$ where $\mathbb{Z}$ is seen as an object of $\ho(\Ch)$ concentrated in degree $0$. \end{proposition} \begin{proof} Consider the commutative square $\begin{tikzcd} \sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\ \sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0). \end{tikzcd}$ It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left vertical morphisms of the above square are isomorphisms. Then, an immediate computation left to the reader 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 and $\sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}$. \end{proof} \begin{remark} 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 slice $\oo$\nbd{}categories (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 oplax contractible. \end{proposition} \begin{proof} This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}. \end{proof} \section{Homology of globes and spheres} \begin{lemma}\label{lemma:globescontractible} For every $n \in \mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is oplax contractible. \end{lemma} \begin{proof} Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}cells for every $k$ such that $0\leq kn$:] Since $\sS_n$ is an $n$\nbd{}category, the image of $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If $x$ is a unit on a cell whose dimension is strictly lower than $n$, then everything works like in the case $k0. \] \emph{From now on, let us use the convention that$\trgt_0(x)=x$when$x$is a$0$\nbd{}cell of$X$}. When$n>0$, the source and target of an$n$\nbd{}cell$(x,p)$of$X/a$are given by $\src((x,p))=(\src(x),p) \text{ and } \trgt((x,p))=(\trgt(x),p).$ %% LA DESCRIPTION DU BUT AU DESSUS N'EST PAS BONNE POUR LA DIMENSION 1 %% A CORRIGER !! Moreover, the$\oo$\nbd{}functor$f/a$is described as $(x,p) \mapsto (f(x),p),$ and the canonical$\oo$\nbd{}functor$X/a \to X$as $(x,p) \mapsto x.$ \end{paragr} \begin{paragr}\label{paragr:unfolding} Let$f : X \to A$be an$\oo$\nbd{}functor with$A$a$1$\nbd{}category. Every arrow$\beta : a \to a'$of$A$induces an$\oo\nbd{}functor \begin{align*} X/\beta : X/a &\to X/{a'} \\ (x,p) & \mapsto (x,\beta \circ p), \end{align*} which takes part in a commutative triangle $\begin{tikzcd}[column sep=tiny] X/{a} \ar[rr,"X/{\beta}"] \ar[dr] && X/{a'} \ar[dl] \\ &X&. \end{tikzcd}$ This defines a functor \begin{align*} X/{-} : A &\to \oo\Cat\\ a &\mapsto X/a \end{align*} and a canonical\oo$\nbd{}functor $\colim_{a \in A} (X/{a}) \to X.$ Let$f' : X' \to A$be another$\oo$\nbd{}functor and let $\begin{tikzcd} X \ar[rr,"g"] \ar[dr,"f"'] && X' \ar[dl,"f'"] \\ &A& \end{tikzcd}$ be a commutative triangle in$\oo\Cat$. Recall from \ref{paragr:slices} that for every object$a$of$A$, there is an$\oo$\nbd{}functor$g/adefined as \begin{align*} g/a : X/a &\to X'/a \\ (x,p) &\mapsto (g(x),p). \end{align*} This defines a natural transformation $g/- : X/- \Rightarrow X'/-,$ and thus induces an\oo$\nbd{}functor $\colim_{a \in A}(X/a) \to \colim_{a \in A}(X'/a).$ Furthermore, it is immediate to check that the square $\begin{tikzcd} \displaystyle \colim_{a \in A}(X/a) \ar[d] \ar[r] & X \ar[d,"g"] \\ \displaystyle\colim_{a \in A}(X'/a) \ar[r] & X', \end{tikzcd}$ is commutative. \end{paragr} \begin{lemma}\label{lemma:colimslice} Let$f : X \to A$be an$\oo$\nbd{}functor such that$A$is a$1$\nbd{}category. The canonical$\oo$\nbd{}functor $\colim_{a \in A}(X/a) \to X$ is an isomorphism. \end{lemma} \begin{proof} We have to show that the cocone $(X/a \to X)_{a \in \Ob(A)}$ is colimiting. Let $(\phi_{a} : X/a \to C)_{a \in \Ob(A)}$ be another cocone and let$x$be a$n$\nbd{}arrow of$X$. Notice that the pair $(x,1_{f(\trgt_0(x))})$ is a$n$\nbd{}arrow of$X/f(\trgt_0(x)). We leave it to the reader to check that the formula \begin{align*} \phi : X &\to C \\ x &\mapsto \phi_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))}). \end{align*} defines an\oo$\nbd{}functor and it is straightforward to check that for every object$a$of$A$the triangle $\begin{tikzcd} X/a\ar[dr,"\phi_{a}"']\ar[r] & X \ar[d,"\phi"] \\ & C \end{tikzcd}$ is commutative. This proves the existence part of the universal property. Now let$\phi' : X \to C$be another$\oo$\nbd{}functor that makes the previous triangles commute for every object$a$of$A$and let$x$be an$n$\nbd{}cell of$X$. Since the triangle $\begin{tikzcd} X/f(\trgt_0(x)) \ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r] & X \ar[d,"\phi'"] \\ & C \end{tikzcd}$ is commutative, we necessarily have $\phi'(x)=\phi_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))})$ which proves that$\phi'=\phi$. \end{proof} \begin{paragr} In particular, when we apply the previous lemma to$\mathrm{id}_A : A \to A$, we obtain that every$1$\nbd{}category$A$is (canonically isomorphic to) the colimit $\colim_{a \in A} (A/a).$ % In other words, this simply say that the colimit of the Yoneda embedding$A \to % \Psh{A}$is the terminal presheaves We now proceed to prove that this colimit is homotopic with respect to the folk weak equivalences. \end{paragr} Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an$\oo$\nbd{}functor$f : X \to A$with$A$a$1$\nbd{}category. \begin{lemma}\label{lemma:sliceisfree} If$X$is free, then for every object$a$of$A$, the$\oo$\nbd{}category$X/a$is free. More precisely, if$\Sigma^X_n$is the$n$\nbd{}basis of$X$, then the$n$\nbd{}basis of$X/a$is the set $\Sigma^{X/a}_n := \{(x,p) \in (X/a)_n \vert x \in \Sigma^X_n \}.$ \end{lemma} \begin{proof} It is immediate to check that for every object$a$of$A$, the canonical forgetful functor$\pi_{a} : A/a \to A$is a discrete Conduché functor (see Section \ref{section:conduche}). Hence, from Lemma \ref{lemma:pullbackconduche} we know that$X/a \to X$is a discrete Conduché$\oo$\nbd{}functor. The result follows then from Theorem \ref{thm:conduche}. \end{proof} \begin{paragr} When$X$is free, every arrow$\beta : a \to a'$of$Ainduces a map \begin{align*} \Sigma^{X/a}_n &\to \Sigma^{X/a'}_n \\ (x,p) &\mapsto (x,\beta\circ p). \end{align*} This defines a functor \begin{align*} \Sigma^{X/{\shortminus}}_n : A &\to \Set \\ a &\mapsto \Sigma^{X/a}_n. \end{align*} \end{paragr} \begin{lemma}\label{lemma:basisofslice} IfX$is free, then there is an isomorphism of functors $\Sigma^{X/\shortminus}_n \simeq \coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),\shortminus\right).$ \end{lemma} \begin{proof} For every object$a$of$A$and every$x \in \Sigma_n^X, we have a canonical map \begin{align*} \Hom_A\left(f(\trgt_0(x)),a\right) &\to \Sigma^{X/a}_n \\ p &\mapsto (x,p). \end{align*} By universal property, this induces a map $\coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),a\right) \to \Sigma^{X/a}_n,$ which is natural ina$. A simple verification shows that it is a bijection. \end{proof} \begin{proposition}\label{prop:sliceiscofibrant} Let$A$be a$1$\nbd{}category,$X$be a free$\oo$\nbd{}category and$f : X \to A$be an$\oo\nbd{}functor. The functor \begin{align*} A &\to \oo\Cat \\ a &\mapsto X/a \end{align*} is a cofibrant object for the projective model structure on\oo\Cat(A)$induced by the folk model structure on$\oo\Cat$(\ref{paragr:projmod}). \end{proposition} \begin{proof} Recall that the set $\{i_n: \sS_{n-1} \to \sD_n \vert n \in \mathbb{N} \}$ is a set of generating folk cofibrations. From Lemmas \ref{lemma:sliceisfree} and \ref{lemma:basisofslice} we deduce that for every object$a$of$A$and every$n \in \mathbb{N}$, the canonical square $\begin{tikzcd} \displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),a)}\sS_{n-1} \ar[r] \ar[d] & \sk_{n-1}(X/a) \ar[d]\\ \displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),a)}\sD_n \ar[r]& \sk_n{(X/a)} \end{tikzcd}$ is cocartesian. It is straightforward to check that this square is natural in$a$in an obvious sense, which means that we have a cocartesian square in$\oo\Cat(A)$: $\begin{tikzcd} \displaystyle\coprod_{x \in \Sigma^X_n}\sS_{n-1}\otimes f(\trgt_0(x)) \ar[r] \ar[d] & \sk_{n-1}(X/-) \ar[d]\\ \displaystyle\coprod_{x \in \Sigma^X_n}\sD_n\otimes f(\trgt_0(x)) \ar[r]& \sk_n{(X/-)} \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd}$ (see \ref{paragr:cofprojms} for notations). From the second part of Proposition \ref{prop:modprs}, we deduce that for every$n\geq 0$, $\sk_{n-1}(X/-) \to \sk_{n}(X/-)$ is a cofibration for the projective model structure on$\oo\Cat(A)$. Thus, the transfinite composition $\emptyset \to \sk_{0}(X/-) \to \sk_{1}(X/) \to \cdots \to \sk_{n}(X/-) \to \cdots,$ which is canonically isomorphic to$\emptyset \to X/-$(see Lemma \ref{lemma:filtration}), is also a cofibration for the projective model structure. \end{proof} \begin{corollary}\label{cor:folkhmtpycol} Let$A$be a$1$\nbd{}category and$f : X \to A$be an$\oo$\nbd{}functor. The canonical arrow of$\ho(\oo\Cat^{\folk})$$\hocolim^{\folk}_{a \in A}(X/a) \to X,$ induced by the co-cone$(X/a \to X)_{a \in \Ob(A)}$, is an isomorphism. \end{corollary} Beware that in the previous corollary, we did \emph{not} suppose that$X$was free. \begin{proof} Let$P$be a free$\omega$-category and$g : P \to X$a folk trivial fibration and consider the following commutative diagram of$\ho(\oo\Cat^{\folk})$$$\label{comsquare} \begin{tikzcd} \displaystyle\hocolim^{\folk}_{a \in A}(P/a) \ar[d] \ar[r] & \displaystyle\colim_{a \in A}(P/a) \ar[d] \ar[r] & P \ar[d]\\ \displaystyle\hocolim^{\folk}_{a \in A}(X/a) \ar[r] & \displaystyle\colim_{a \in A}(X/a) \ar[r] & X \end{tikzcd}$$ where the middle and most left vertical arrows are induced by the arrows $g/a : P/a \to X/a,$ and the most right vertical arrow is induced by$g$. Since trivial fibrations are stable by pullback,$g/a$is a trivial fibration. This proves that the most left vertical arrow of diagram \eqref{comsquare} is an isomorphism. Now, from Proposition \ref{prop:sliceiscofibrant} and Corollary \ref{cor:cofprojms}, we deduce that the arrow $\hocolim_{a \in A}^{\folk}(P/a)\to \colim_{a \in A}(P/a)$ is an isomorphism. Moreover, from Lemma \ref{lemma:colimslice}, we know that the arrows $\colim_{a \in A}(P/a)\to P$ and $\colim_{a \in A}(X/a)\to X$ are isomorphisms. Finally, since$g$is a folk weak equivalence, the most right vertical arrow of diagram \eqref{comsquare} is an isomorphism and by an immediate 2-out-of-3 property this proves that all arrows of \eqref{comsquare} are isomorphisms. In particular, so is the composition of the two bottom horizontal arrows, which is what we desired to show. \end{proof} We now move on to the next step needed to prove that every$1$\nbd{}category is \good{}. For that purpose, let us recall a construction commonly referred to as the Grothendieck construction''. \begin{paragr} Let$A$be a$1$\nbd{}category and$F : A \to \Cat$a functor. We denote by$\int F$or$\int_{a \in A}F(a)$the category such that: \begin{itemize}[label=-] \item An object of$\int F$is a pair$(a,x)$where$a$is an object of$A$and$x$is an object of$F(a)$. \item An arrow$(a,x) \to (a',x')$of$\int F$is a pair$(f,k)$where $f : a \to a'$ is an arrow of$A$, and $k : F(f)(x) \to x'.$ \end{itemize} The unit on$(a,x)$is the pair$(1_a,1_x)$and the composition of$(f,k) : (a,x) \to (a',x')$and$(f',k') : (a',x') \to (a'',x'')is given by: $(f',k')\circ(f,k)=(f'\circ f,k'\circ F(f')(k)).$ Every natural transformation $\begin{tikzcd} A \ar[r,bend left,"F",""{name=A,below},pos=19/30]\ar[r,bend right,"G"',""{name=B, above},pos=11/20] & \Cat \ar[from=A,to=B,Rightarrow,"\alpha",pos=9/20] \end{tikzcd}$ induces a functor \begin{align*} \int \alpha : \int F &\to \int G\\ (a,x) &\mapsto (a,\alpha_a(x)). \end{align*} Altogether, this defines a functor \begin{align*} \int : \Cat(A)&\to \Cat \\ F&\mapsto \int F, \end{align*} where\Cat(A)$is the category of functors from$A$to$\Cat$. \end{paragr} We now recall an important theorem due to Thomason. \begin{theorem}[Thomason]\label{thm:Thomason} The functor$\int : \Cat(A) \to \Cat$sends the pointwise Thomason equivalences (\ref{paragr:homder}) to Thomason equivalences and the induced functor $\overline{\int} : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th})$ is canonically isomorphic to the homotopy colimit functor $\hocolim^{\Th}_A : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th}).$ \end{theorem} \begin{proof} The original source for this Theorem is \cite{thomason1979homotopy}. However, the definition of homotopy colimit used by Thomason, albeit equivalent, is not the same as the one we used in this dissertation and is slightly outdated. A more modern proof of the theorem can be found in \cite[Proposition 2.3.1 and Théorème 1.3.7]{maltsiniotis2005theorie}. \end{proof} \begin{corollary}\label{cor:thomhmtpycol} Let$A$be a$1$\nbd{}category. The canonical map $\hocolim^{\Th}_{a \in A}(A/a) \to A$ induced by the co-cone$(A/a \to A)_{a \in \Ob(A)}$, is an isomorphism of$\ho(\Cat^{\Th})$. \end{corollary} \begin{proof} For every object$a$of$A$, the canonical map to the terminal category $A/a \to \sD_0$ is a Thomason equivalence. This comes from the fact that$A/a$is oplax contractible (Proposition \ref{prop:slicecontractible}), or from \cite[Section 1, Corollary 2]{quillen1973higher} and the fact that$A/a$has a terminal object. In particular, the morphism of functors $A/(-) \Rightarrow k_{\sD_0},$ where$k_{\sD_0}$is the constant functor$A \to \Cat$with value the terminal category$\sD_0$, is a pointwise Thomason equivalence. It follows from the first part of Theorem \ref{thm:Thomason} that $\int_{a \in A}A/a \to \int_{a \in A}k_{\sD_0}$ is a Thomason equivalence and an immediate computation shows that $\int_{a \in A}k_{\sD_0} \simeq A.$ From the second part of Theorem \ref{thm:Thomason}, we have that $\hocolim^{\Th}_{a \in A}(A/a) \simeq A.$ A thorough analysis of all the isomorphisms involved shows that this last isomorphism is indeed induced by the co-cone$(A/a \to A)_{a \in \Ob(A)}$. \end{proof} \begin{remark} It is possible to extend the previous corollary to prove that for every functor$f : X \to A$($X$and$A$being$1$\nbd{}categories), we have $\hocolim^{\Th}_{a \in A} (X/a) \simeq X.$ However, to prove that it is also the case when$X$is an$\oo$\nbd{}category and$f$an$\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in$\oo\Cat$and to prove an$\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation. \end{remark} Putting all the pieces together, we are now able to prove the awaited Theorem. \begin{theorem}\label{thm:categoriesaregood} Every$1$\nbd{}category is \good{}. \end{theorem} \begin{proof} All the arguments of the proof have already been given and we sum them up here essentially for the sake of clarity. Let$A$be a$1\nbd{}category. Consider the diagram \begin{align*} A &\to \oo\Cat\\ a &\mapsto A/a \end{align*} and the co-cone $(A/a \to A)_{a \in \Ob(A)}.$ \begin{itemize}[label=-] \item The canonical map of\ho(\oo\Cat^{\folk})$$\hocolim_{a \in A}^{\folk} (A/a) \to A$ is an isomorphism thanks to Corollary \ref{cor:folkhmtpycol} applied to$\mathrm{id}_A : A \to A$. \item The canonical map of$\ho(\oo\Cat^{\Th})$$\hocolim_{a \in A}^{\Th} (A/a) \to A$ is an isomorphism thanks to Corollary \ref{cor:thomhmtpycol} and the fact that the canonical morphisms of op-prederivators$\Ho(\Cat^{\Th}) \to \Ho(\oo\Cat^{\Th})$is homotopy cocontinuous (see \ref{paragr:thomhmtpycol}). \item Every$A/a$is \good{} thanks to Proposition \ref{prop:contractibleisgood} and Proposition \ref{prop:slicecontractible}. \end{itemize} Thus, Proposition \ref{prop:criteriongoodcat} applies and this proves that$A\$ is \good{}. \end{proof} %%% Local Variables: %%% mode: latex %%% TeX-master: "main" %%% End: