### Beaucoup de typos corrigés. Il faudrait relire la preuve du Lemme 4.6.5

parent 02a5625a
 ... ... @@ -415,7 +415,27 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow] \end{tikzcd} \] \end{paragr} Moreover, the functor $\lambda$ being left Quillen, it is strongly derivable (Definition \ref{def:strnglyder}) and hence induces a morphism of op-prederivators, which we again denote by $\sH^{\pol}$: $\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch).$ % We also have a universal $2$\nbd{}morphism which we again denote by $\alpha^{\pol}$: % $% \begin{tikzcd} % \oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ % \Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\pol}"'] & \Ho(\Ch). % \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow] % \end{tikzcd} %$ \end{paragr} The following proposition is an immediate consequence of Theorem \ref{thm:cisinskiII}. \begin{proposition}\label{prop:polhmlgycocontinuous} The polygraphic homology $\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)$ is homotopy cocontinuous. \end{proposition} As we shall now see, oplax homotopy equivalences (Definition \ref{def:oplaxhmtpyequiv}) induce isomorphisms in polygraphic homology. In order to prove that, we first need a couple of technical lemmas. \begin{lemma}\label{lemma:liftingoplax} Let ... ... @@ -433,7 +453,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c \] \end{lemma} \begin{proof} Notice first that because of the natural isomorphism $(\sD_0\amalg \sD_0) \otimes C \simeq C \amalg C$ we have that $\alpha : f_0 \Rightarrow f_1$ can be encoded in a functor $\alpha : \sD_1 \otimes C \to D$ such that the diagram Notice first that because of the natural isomorphism $(\sD_0\amalg \sD_0) \otimes C \simeq C \amalg C,$ we have that $\alpha : f_0 \Rightarrow f_1$ can be encoded in a functor $\alpha : \sD_1 \otimes C \to D$ such that the diagram $\begin{tikzcd} (\sD_0\amalg \sD_0) \otimes C \simeq C \amalg C \ar[d,"i_1 \otimes C"'] \ar[dr,"{\langle u, v \rangle}"] &\\ ... ... @@ -479,7 +499,7 @@ From now on, for an \oo\nbd{}functor u, we write \sH^{\pol}(u) instead of \[ q : D' \to D$ be trivial fibrations for the canonical model structure with $C'$ and $D'$ cofibrant. Using that $q$ is a trivial fibration and $C'$ is cofibrant, we know that there exists $u' : C' \to D'$ and $v' : C' \to D'$ such that the squares be folk trivial fibrations with $C'$ and $D'$ cofibrant. Using that $q$ is a trivial fibration and $C'$ is cofibrant, we know that there exist $u' : C' \to D'$ and $v' : C' \to D'$ such that the squares $\begin{tikzcd} C' \ar[d,"p"] \ar[r,"u'"] & D' \ar[d,"q"] \\ ... ... @@ -510,38 +530,19 @@ The following proposition is an immediate consequence of the previous lemma. Lemma \ref{lemma:liftingoplax}, Lemma \ref{lemma:oplaxpolhmlgy} and Proposition \ref{prop:oplaxhmtpypolhmlgy} are also true if we replace oplax'' by lax'' everywhere. \end{remark} \begin{paragr} The functor \lambda being left Quillen, it is strongly derivable (Definition \ref{def:strnglyder}) and hence also induces a morphism of op-prederivators, which we again denote by \sH^{\pol}: \[ \sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch).$ Moreover, we also have a universal $2$-morphism which we again denote by $\alpha^{\pol}$: $\begin{tikzcd} \oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\pol}"'] & \Ho(\Ch). \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow] \end{tikzcd}$ \end{paragr} The following proposition is an immediate consequence of Theorem \ref{thm:cisinskiII}. \begin{proposition}\label{prop:polhmlgycocontinuous} The polygraphic homology $\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)$ is homotopy cocontinuous. \end{proposition} \section{Singular homology as derived abelianization}\label{section:singhmlgyderived} We have seen in the previous section that the polygraphic homology functor is the total left derived functor of $\lambda : \oo\Cat \to \Ch$ when $\oo\Cat$ is equipped with the folk weak equivalences. As it turns out, the abelianization functor is also totally left derivable when $\oo\Cat$ is equipped with the Thomason equivalences and the total left derived functor is the singular homology functor. In order to prove this result, we first need a few technical lemmas. \begin{lemma}\label{lemma:nuhomotopical} Let $\nu : \Ch \to \oo\Cat$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason equivalences. Let $\nu : \Ch \to \oo\Cat$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends the quasi-isomorphisms to Thomason equivalences. \end{lemma} \begin{proof} We have already seen that $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$. By adjunction, this means that $\nu$ is right Quillen for this model structure. In particular, it sends trivial fibrations of chain complexes to folk trivial fibrations. From Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} and the fact that all chain complexes are fibrant, it follows that $\nu$ sends weak equivalences of chain complexes to weak equivalences of the folk model structure, which are in particular Thomason equivalences (Lemma \ref{lemma:nervehomotopical}). \end{proof} \begin{remark} The proof of the previous lemma shows the stronger result that $\nu$ sends weak equivalences of chain complexes to weak equivalences for the folk model structure on $\oo\Cat$. This will be of no use in the sequel. The proof of the previous lemma shows the stronger result that $\nu$ sends the quasi-ismorphisms to folk weak equivalences. This will be of no use in the sequel. \end{remark} Recall that we write $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ for the left adjoint of the nerve functor ${N_{\oo} : \oo\Cat \to \Psh{\Delta}}$ (see Paragraph \ref{paragr:nerve}). \begin{lemma}\label{lemma:abelor} ... ... @@ -557,7 +558,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins \begin{proof} All the functors involved are cocontinuous, hence it suffices to prove that the triangle is commutative when pre-composed by the Yoneda embedding $\Delta \to \Psh{\Delta}$. This follows immediately from the description of the orientals in \cite{steiner2004omega}. \end{proof} Recall now that the notions of adjunction and equivalence are valid in every $2$-category and in particular in the $2$\nbd{}category of pre-derivators (see \ref{paragr:prederequivadjun}). We omit the proof of the following lemma, which is the same as when the ambient $2$-category is the $2$-category of categories. Recall now that the notions of adjunction and equivalence are valid in every $2$\nbd{}category and in particular in the $2$\nbd{}category of pre-derivators (see \ref{paragr:prederequivadjun}). We omit the proof of the following lemma, which is the same as when the ambient $2$\nbd{}category is the $2$\nbd{}category of categories. \begin{lemma}\label{lemma:adjeq} 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} ... ... @@ -613,27 +614,34 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins From Proposition \ref{prop:gonzalezcritder}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $\LL\lambda^{\Th} \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the singular homology. \end{proof} \begin{remark} Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ sends all weak equivalences of simplicial sets to Thomason equivalences nor $\lambda : \oo\Cat \to \Ch$ sends all Thomason equivalences to quasi-isomorphisms. But this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does send all weak equivalences of simplicial sets to quasi-isomorphisms. Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ sends all weak equivalences of simplicial sets to Thomason equivalences nor $\lambda : \oo\Cat \to \Ch$ sends all Thomason equivalences to quasi-isomorphisms. However, this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does send all weak equivalences of simplicial sets to quasi-isomorphisms. \end{remark} \begin{paragr}\label{paragr:univmor} Since $\sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)$ is the left derived morphisms of op-prederivators of the abelianization functor, it comes with a universal $2$-morphism Since $\sH^{\sing} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)$ is the left derived functor of the abelianization functor, it comes with a universal natural transformation $\begin{tikzcd} \oo\Cat \ar[d,"\gamma^{\Th}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch). \ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \ho(\Ch). \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 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 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 a natural transformation $\lambda c_{\oo} N_{\oo} \Rightarrow \lambda.$ Then $\alpha^{\sing}$ is nothing but the following composition of $2$\nbd{}morphisms Then $\alpha^{\sing}$ is nothing but the following composition of natural transformations $\begin{tikzcd}[column sep=huge] \oo\Cat \ar[d,"\gamma^{\Th}"]\ar[r,bend left,"\lambda",""{name=A,below}] \ar[r,"\lambda c_{\oo} N_{\oo}"',""{name=B,above}] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch), \ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \ho(\Ch), \ar[from=B,to=A,Rightarrow]\ar[from=1-1,to=2-2,phantom,"\simeq" description] \end{tikzcd}$ ... ... @@ -691,7 +699,7 @@ Another consequence of the above counter-example is the following result, which Suppose the converse, which is that the functor $\sH^{\pol} \circ \gamma^{\folk} : \oo\Cat \to \ho(\Ch)$sends Thomason equivalences to isomorphisms of $\ho(\Ch)$. Because of the inclusion $\W^{\folk} \subseteq \W^{\Th}_{\oo}$, the category $\ho(\oo\Cat^{\Th})$ may be identified with the localization of $\ho(\oo\Cat^{\folk})$ with respect to $\gamma^{\folk}(\W^{\Th}_{\oo})$ and then the localization functor is nothing but \]sends the Thomason equivalences to isomorphisms of $\ho(\Ch)$. Because of the inclusion $\W^{\folk} \subseteq \W^{\Th}_{\oo}$, the category $\ho(\oo\Cat^{\Th})$ may be identified with the localization of $\ho(\oo\Cat^{\folk})$ with respect to $\gamma^{\folk}(\W^{\Th}_{\oo})$ and then the localization functor is nothing but $\J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).$ ... ... @@ -761,7 +769,7 @@ Another consequence of the above counter-example is the following result, which that \emph{singular homology} is the only correct'' homology of $\oo$\nbd{}categories. \end{remark} \begin{paragr}\label{paragr:defcancompmap} Even though triangle \eqref{cmprisontrngle} is not commutative (even up to an isomorphism), it can be filled up with a $2$-morphism. Indeed, consider the following $2$\nbd{}square Even though triangle \eqref{cmprisontrngle} is not commutative (even up to an isomorphism), it can be filled up with a $2$\nbd{}morphism. Indeed, consider the following $2$\nbd{}square $\begin{tikzcd} \oo\Cat \ar[d,"\gamma^{\Th}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ ... ... @@ -769,7 +777,9 @@ Another consequence of the above counter-example is the following result, which \ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow] \end{tikzcd}$ Since $\gamma^{\Th}=\J\circ \gamma^{\folk}$ and the polygraphic homology is the total left derived functor of the abelianization functor when $\oo\Cat$ is equipped with folk weak equivalences, we obtain by universal property (see \ref{paragr:defleftderived}) a unique natural transformation Since $\gamma^{\Th}=\J\circ \gamma^{\folk}$ and the polygraphic homology is the total left derived functor of the abelianization functor when $\oo\Cat$ is equipped with the folk weak equivalences, we obtain by universal property (see \ref{paragr:defleftderived}) a unique natural transformation \begin{equation}\label{cmparisonmapdiag} \begin{tikzcd} \ho(\oo\Cat^{\folk}) \ar[d,"\J"] \ar[rd,"\sH^{\pol}",""{name=A,below}] & \\ ... ... @@ -958,8 +968,8 @@ The previous proposition admits the following corollary, which will be of great \begin{proposition}\label{prop:fmsncat} There exists a model structure on $n\Cat$ such that: \begin{itemize}[label=-] \item weak equivalences are exactly those morphisms $f : C \to D$ such that $\iota_n(f)$ is a weak equivalence for the folk model structure on $\oo\Cat$, \item fibrations are exactly those morphisms $f : C \to D$ such that $\iota_n(f)$ is a fibrations for the folk model structure on $\oo\Cat$. \item the weak equivalences are exactly those morphisms $f : C \to D$ such that $\iota_n(f)$ is a weak equivalence for the folk model structure on $\oo\Cat$, \item the fibrations are exactly those morphisms $f : C \to D$ such that $\iota_n(f)$ is a fibrations for the folk model structure on $\oo\Cat$. \end{itemize} Moreover, there exists a set $I$ of generating cofibrations (resp.\ a set $J$ of generating trivial cofibrations) for the folk model structure on $\oo\Cat$ such that the image by $\tau^{i}_{\leq n}$ of $I$ (resp.\ $J$) is a set of generating cofibrations (resp.\ generating trivial cofibrations) of the above model structure on $n\Cat$. \end{proposition} ... ... @@ -1025,18 +1035,18 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati exists a $0$\nbd{}cell $x'$ of $C$ such that $f(x')=y'$ and then if we set $x:=\eta_C(x')$, we have $T(f)(x)=y$. Now let $x,y$ be parallel $k$\nbd{}cells of $T(C)$ and suppose given a $(k+1)$\nbd{}cell $\beta : f(x) \to f(y)$ of $T(D)$. We have to distinguish Now let $x,y$ be parallel $k$\nbd{}cells of $T(C)$ and let $\beta : f(x) \to f(y)$ be a $(k+1)$\nbd{}cell of $T(D)$. We have to distinguish several cases. \begin{description} \item[Case $kn$ ... ... @@ -1050,13 +1060,13 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati $(z_0,\beta_1,z_1,\cdots,z_{p-1},\beta_p,z_p)$ where the $z_i$ are all parallel $n$\nbd{}cells of $C$ with $z_0=f(x')$ and $z_p=f(y')$, and each $\beta_i$ is $(n+1)$\nbd{}cell of $C$ either where the $z_i$ are all parallel $n$\nbd{}cells of $D$ with $z_0=f(x')$ and $z_p=f(y')$, and each $\beta_i$ is $(n+1)$\nbd{}cell of $D$ either from $z_{i-1}$ to $z_i$ or from $z_{i}$ to $z_{i-1}$. Using the fact that $f$ is a folk trivial fibration, it is easy to prove the existence of a zigzag from $x'$to $y'$, which implies in particular that $x=\eta_C(x')=\eta_C(y')=y$. \item[Case $k>n$:] Since all $k$\nbd{}cells of $T(C)$ and $T(D)$ with $k>n$ are units, we trivially have $f(x)=f(y)$ and $x=y$. are units, we trivially have $f(x)=f(y)$ (and $\beta$ is the unit on $f(x)$) and $x=y$. \end{description} Altogether, this proves that $T(f)$ is a folk trivial fibration, hence a folk weak equivalence. ... ... @@ -1068,23 +1078,49 @@ For later reference, we put here the following lemma. \begin{proof} Since $\tau^{i}_{\leq n}$ is a left Quillen functor, it suffices in virtue of Proposition \ref{prop:freeiscofibrant} to show that there exists a free $\oo$\nbd{}category $C'$ such that $\tau^{i}_{\leq n}(C')=C$. Let us write $\Sigma_k$ for the $k$\nbd{}base of $C$ with $0\leq k\leq n-1$ and let $C'$ be the free $\oo$\nbd{}category such that: \begin{itemize}[label=-] \item the $k$\nbd{}base of $C'$ is $\Sigma_k$ for every $0 \leq k \leq n-1$, \item the $n$\nbd{}base of $C'$ is the set $C_n$, \item the $(n+1)$\nbd{}base of $C'$ is the set $\{(x,y)\,\vert\, x \text{ and } y \text{ are parallel } n \text{-cells of } C\},$ the source (resp.\ target) of $(x,y)$ being $x$ (resp.\ $y$), \item the $k$\nbd{}base of $C'$ is empty for $k > n+1$ (i.e.\ $C'$ is an $(n+1)$\nbd{}category). \end{itemize} (For such recursive constructions of free $\oo$\nbd{}categories, see Section \ref{section:freeoocataspolygraph}, and in particular Proposition \ref{prop:freeonpolygraph}.) We invite the reader to verify for himself that indeed $\tau^{i}_{\leq n}(C')=C$. there exists a free $\oo$\nbd{}category $C'$ such that $\tau^{i}_{\leq n}(C')=C$. % In the case $n=0$, this is trivial since it suffices to take % $C'=\iota_0(C)$ (i.e. the set $C$ considered as an $\oo$\nbd{}category, which is % trivially cofibrant). Suppose now that $n>0$. First, consider the $n$\nbd{}category $(U_{n-1}(C))^*$ (for the notations, see \ref{paragr:freecext} and \ref{paragr:cextlowdimension}). This $n$\nbd{}category has the same $k$\nbd{}cells as $C$ for $k n+1$ (i.e.\ $C'$ is an $(n+1)$\nbd{}category). % \end{itemize} % (For such recursive constructions of free $\oo$\nbd{}categories, see Section % \ref{section:freeoocataspolygraph}, and in particular Proposition % \ref{prop:freeonpolygraph}.) % We invite the reader to verify for himself that indeed $\tau^{i}_{\leq n}(C')=C$. \end{proof} \begin{example} Every (small) category is cofibrant for the folk model structure on $\Cat$. ... ... @@ -1095,7 +1131,8 @@ We now turn to truncations of chain complexes. $K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_n,$ where $\partial \circ \partial =0$, and morphisms of $\Ch^{\leq n}$ are defined the expected way. Write $\iota_n : \Ch^{\leq n} \to \Ch$ for the canonical functor that sends an object $K$ of $\Ch^{\leq n}$ to the chain complex where $\partial \circ \partial =0$, and morphisms of $\Ch^{\leq n}$ are defined the expected way. We write $\iota_n : \Ch^{\leq n} \to \Ch$ for the canonical functor that sends an object $K$ of $\Ch^{\leq n}$ to the chain complex $K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_n \longleftarrow 0 \longleftarrow 0 \longleftarrow \cdots.$ ... ... @@ -1406,7 +1443,7 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t $$, $$ and $$ and let $i : \Delta_{\leq 2} \to \Delta$ be the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* : \Psh{\Delta_{\leq 2}} \to \Psh{\Delta}$. Recall that the nerve of a (small) category is $2$-coskeletal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have \Psh{\Delta_{\leq 2}} \to \Psh{\Delta}$. Recall that the nerve of a (small) category is$2$\nbd{}coskeletal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category$D$, the unit morphism$ N_1(D) \to i_* i^* (N_1(D))is an isomorphism of simplicial sets. In particular, we have \begin{align*} \Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)) &\simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),i_* i^* (N_1(D)))\\ &\simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))). ... ... No preview for this file type  ... ... @@ -724,7 +724,7 @@ We can now prove the following proposition, which is the key result of this sect Conversely, ifE$is an$(n+1)$\nbd{}base of$C$, then we can define an$(n+1)$\nbd{}functor$C \to \E_E^*$that sends$E$, seen as a subset of$C_{n+1}$, to$E$, seen as a subset of$(\E^*_E)_{n+1}$(and which is obviously the identity on cells of dimension strictly lower than$n+1$). The fact that$C$and$\E^*$have$E$as an$(n+1)$\nbd{}base implies that this$(n+1)$\nbd{}functor$C \to \E^*$is the inverse of the canonical one$\E^* \to C$. \end{proof} \begin{paragr} \begin{paragr}\label{paragr:cextlowdimension} We extend the definitions and the results from \ref{def:cellularextension} to \ref{prop:criterionnbasis} to the case$n=-1$by saying that a$(-1)$-cellular extension is simply a set$\Sigma$(which is the set of indeterminates) and$(-1)\Cat^+$is the category of sets. Since a$0\Cat\$ is also the category of sets, it makes sense to define the functors ... ...
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