Commit 97f10611 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

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 $k<n-1$:] Since $\eta_C$ and $\eta_D$ are identities on $k$-cells for every $0\leq k<n$ and since $f$ is a folk trivial fibration, there exists a $k$\nbd{}cell $\alpha : x \to y$ of $T(C)$ such that
\item[Case $k<n-1$:] Since $\eta_C$ and $\eta_D$ are identities on $k$-cells for every $0\leq k<n$ and since $f$ is a folk trivial fibration, there exists a $(k+1)$\nbd{}cell $\alpha : x \to y$ of $T(C)$ such that
\[
T(f)(\alpha)=\beta.
\]
\item[Case $k=n-1$:] By definition of $T(D)$, there exists an $n$\nbd{}cell
$\beta ' : f(x) \to f(y)$ of $D$ such that
$\eta_{D}(\beta')=\beta$. Since $f$ is a folk trivial fibration, there
exists a $n$\nbd{}cell $\alpha' : x \to y$ of $C$ such that
exists an $n$\nbd{}cell $\alpha' : x \to y$ of $C$ such that
$f(\alpha')=\beta'$. If we set $\alpha:=\eta_{C}(\alpha')$, we have
$T(f)(\alpha)=\beta$.
\item[Case $k=n$:] Since all $l$\nbd{}cells of $T(C)$ and $T(D)$ with $l>n$
......@@ -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 $k<n$ and has exactly one
\emph{generating} $n$\nbd{}cell for each $n$\nbd{}cell of $C$. It is obviously free and we have a canonical $n$\nbd{}functor
\[
\epsilon_C : (U_{n-1}(C))^* \to C,
\]
given by the co-unit of the adjunction $(-)^* \dashv U_{n-1}(-)$. Now, let $C'$
be the $(n+1)$\nbd{}category (considered as an $\oo$\nbd{}category) that has the same $k$\nbd{}cells as
$(U_{n-1}(C))^*$ for $k\leq n$ and whose set of $(n+1)$\nbd{}cells is freely
generated be the set
\[
\{(x,y)\,\vert\, x \text{ and } y \text{ are parallel } n \text{-cells of }
(U_{n-1}(C))^* \text{ such that }\epsilon_C(x)=\epsilon_C(y)\}.
\]
The $(n+1)$\nbd{}category $C'$ is obviously free and it is a harmless
verification, which we leave to reader, to check that $\tau^i_{\leq n}(C')=C$.
% \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 $C_n$,
% \end{itemize}
% By universal property of $n$\nbd{}bases (see \ref{}, \ref{} and \ref{}), there a unique map
% $\rho : C'_n \to C_n$ such that $\rho(x)=x$ for every $x \in C_n$ and
% \[
% \rho(x\comp_ky)=\rho(x)\comp_k\rho(y)
% \]
% for all $x,y \in C'_n$ that are $k$\nbd{}composable.
% \begin{itemize}[label=-]
% \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 $\tau^{i}_{\leq n}(C')=C$.
\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
$[0]$, $[1]$ and $[2]$ 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))).
......
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......@@ -724,7 +724,7 @@ We can now prove the following proposition, which is the key result of this sect
Conversely, if $E$ 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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