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 $\alpha^{\pol}$:

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 $\alpha^{\pol}$:

\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

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

\]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

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

@@ -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 $\oo\Cat$.

\itemthe 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 $\oo\Cat$,

\itemthe 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.

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

@@ -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 $i : \Delta_{\leq2}\to\Delta$ be the

canonical inclusion. This inclusion induces by pre-composition a functor $i^*

: \Psh{\Delta}\to\Psh{\Delta_{\leq2}}$ which has a right-adjoint $i_* :

\Psh{\Delta_{\leq2}}\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_{\leq2}}\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

@@ -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