\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 generated 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, known as the \emph{projective model structure on $\Ch$}, 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 cofibrations are the morphisms of chain complexes $f: X\to Y$ such that for every $n\geq0$, $f_n : X_n \to Y_n$ is a monomorphism with projective cokernel,

...

...

@@ -23,7 +23,7 @@ We denote by $\Ch$ the category of non-negatively graded chain complexes of abel

in the expected way.

\end{paragr}

\begin{paragr}

Recall that an $n$-simplex $x$ of a simplicial set $X$ is \emph{degenerate} if there exists an epimorphism $\varphi : [n]\to[m]$ with $m<n$ and an $m$-simplex $y$ such that $X(\varphi)(y)=x$. We denote by $D_n(X)$ the subgroup of $K_n(X)$ generated by the degenerate $n$-simplices. We denote by $\kappa_n(X)$ the abelian group of \emph{normalized chain complex},

Recall that an $n$-simplex $x$ of a simplicial set $X$ is \emph{degenerate} if there exists an epimorphism $\varphi : [n]\to[m]$ with $m<n$ and an $m$-simplex $y$ such that $X(\varphi)(y)=x$. We denote by $D_n(X)$ the subgroup of $K_n(X)$ generated by the degenerate $n$-simplices and by $\kappa_n(X)$ the abelian group of \emph{normalized chain complex}:

\[

\kappa_n(X)=K_n(X)/D_n(X).

\]

...

...

@@ -38,7 +38,7 @@ This defines a chain complex $\kappa(X)$, which we call the \emph{normalized cha

\end{align*}

\end{paragr}

\begin{lemma}\label{lemma:normcompquil}

The functor $\kappa : \Psh{\Delta}\to\Ch$ is left Quillen and sends weak equivalences of simplicial sets to quasi-isomorphisms.

The functor $\kappa : \Psh{\Delta}\to\Ch$ is left Quillen and sends the weak equivalences of simplicial sets to quasi-isomorphisms.

\end{lemma}

\begin{proof}

Recall that the Quillen model structure on simplicial sets admits the set of inclusions

...

...

@@ -74,7 +74,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

where $H_k : \ho(\Ch)\to\Ab$ is the usual functor that associate to an object of $\ho(\Ch)$ its $k$\nbd{}th homology group.

where $H_k : \ho(\Ch)\to\Ab$ is the usual functor that associates to an object of $\ho(\Ch)$ its $k$\nbd{}th homology group.

\end{paragr}

%% \begin{paragr}

...

...

@@ -303,7 +303,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

\begin{proof}

For $n \geq0$, write $\phi_n : \mathbb{Z}\Sigma_n \to\lambda_n(C)$ for the map defined in \ref{paragr:abelpolmap} (which we know is an isomorphism from Lemma \ref{lemma:abelpol}).

The map $w_n: C_n \to\mathbb{Z}\Sigma_n$ induces a map $\mathbb{Z}C_n \to\mathbb{Z}\Sigma$ by linearity, which in turn induces a map $\lambda_n(C)\to\mathbb{Z}\Sigma_n$ (because $w_n(x \comp_k y)= w_n(x)+w_n(y)$ for every pair $(x,y)$ of $k$\nbd{}composable $n$\nbd{}cells). Write $\psi_n$ for this last map. It is immediate to check that the composition

The map $w_n: C_n \to\mathbb{Z}\Sigma_n$ induces a map $\mathbb{Z}C_n \to\mathbb{Z}\Sigma_n$ by linearity, which in turn induces a map $\lambda_n(C)\to\mathbb{Z}\Sigma_n$ (because $w_n(x \comp_k y)= w_n(x)+w_n(y)$ for every pair $(x,y)$ of $k$\nbd{}composable $n$\nbd{}cells). Write $\psi_n$ for this last map. It is immediate to check that the composition

@@ -311,7 +311,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

Now, for $n>0$, notice that the map $\partial : \mathbb{Z}\Sigma_n \to\mathbb{Z}\Sigma_{n-1}$ given in the statement of the proposition is nothing but the composition

The first part of the proposition follows then from Lemma \ref{lemma:abelpol}.

...

...

@@ -365,13 +365,11 @@ For the definition of \emph{homotopy of chain complexes} see for example \cite[D

is the total left derived functor of $\lambda : \oo\Cat\to\Ch$ (where $\oo\Cat$ is equipped with folk weak equivalences). For an $\oo$\nbd{}category $C$, $\sH^{\pol}(C)$ is the \emph{polygraphic homology of $C$}.

\end{definition}

\begin{paragr}

Similarly to singular homology groups, for $k\geq0$ the $k$\nbd{}th polygraphic homology group of $C$ is defined as

Similarly to singular homology groups, for $k\geq0$ the $k$\nbd{}th polygraphic homology group of an $\oo$\nbd{}category $C$ is defined as

\[

H^{\pol}_k(C):=H_k(\sH^{\pol}(C))

\]

where $H_k : \ho(\Ch)\to\Ab$ is the usual functor that associate to an object of $\ho(\Ch)$ its $k$-th homology group.

In practice, this means that one has to find a cofibrant replacement of $C$, that is to say a free $\oo$\nbd{}category $P$ and a folk trivial fibration

where $H_k : \ho(\Ch)\to\Ab$ is the usual functor that associate to an object of $\ho(\Ch)$ its $k$-th homology group. In practice, this means that one has to find a cofibrant replacement of $C$, that is to say a free $\oo$\nbd{}category $P$ and a folk trivial fibration

\[

P \to C,

\]

...

...

@@ -387,7 +385,7 @@ For the definition of \emph{homotopy of chain complexes} see for example \cite[D

\end{tikzcd}

\]

\end{paragr}

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.

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

\[

...

...

@@ -427,7 +425,7 @@ For the definition of \emph{homotopy of chain complexes} see for example \cite[D

The commutativity of the two induced triangles shows what we needed to prove.

\end{proof}

From now on, when given an $\oo$\nbd{}functor $u$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$ (where $\gamma^{\folk}$ is the localization functor $\oo\Cat\to\ho(\oo\Cat^{\folk})$) for the morphism induced by $u$ at the level of polygraphic homology.

\begin{lemma}

\begin{lemma}\label{lemma:oplaxpolhmlgy}

Let $u,v : C \to D$ two $\oo$\nbd{}functors. If there exists an oplax transformation $u\Rightarrow v$, then

\[

\sH^{\pol}(u)=\sH^{\pol}(v).

...

...

@@ -472,7 +470,7 @@ The following proposition is an immediate consequence of the previous lemma.

Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder whether it is true that \emph{every} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later (Proposition \ref{prop:polhmlgynotinvariant}), it is not the case.

\end{paragr}

\begin{remark}

Proposition \ref{prop:oplaxhmtpypolhmlgy}is also true if we replace ``oplax'' by ``lax''.

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}

...

...

@@ -498,9 +496,9 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

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 folk weak equivalences. As it turns out, the abelianization functor is also totally left derivable when $\oo\Cat$ is equipped with 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.

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 to 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 weak equivalences of chain complexes 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}).

...

...

@@ -522,7 +520,7 @@ 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 notion of adjunction and equivalence is 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 $2$-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.

\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}

...

...

@@ -578,7 +576,7 @@ 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 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 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. 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.

\end{remark}

\begin{paragr}\label{paragr:univmor}

Since $\sH^{\sing} : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch)$ is the left derived of the abelianization functor, it comes with a universal $2$-morphism

...

...

@@ -596,7 +594,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

Then $\alpha^{\sing}$ is nothing but the following composition of $2$\nbd{}morphisms

@@ -631,7 +629,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

\[

C:=B^2\mathbb{N}

\]

(see \ref{paragr:suspmonoid}). As usual, we consider $C$ as an $\oo$\nbd{}category with only unit cells strictly above dimension $2$. This $\oo$\nbd{}category is free; namely its $k$\nbd{}basis is a singleton for $k=0$or $2$ and the empty set otherwise. In particular $C$ is cofibrant for the folk model structure (Proposition \ref{prop:freeiscofibrant}) and it follows from Proposition \ref{prop:abelianizationfreeoocat} that $\sH^{\pol}(C)$ is given by the chain complex (seen as an object of $\ho(\Ch)$)

(see \ref{paragr:suspmonoid}). As usual, we consider $C$ as an $\oo$\nbd{}category with only unit cells strictly above dimension $2$. This $\oo$\nbd{}category is free; namely its $k$\nbd{}basis is a singleton for $k=0$and $k=2$, and the empty set otherwise. In particular $C$ is cofibrant for the folk model structure (Proposition \ref{prop:freeiscofibrant}) and it follows from Proposition \ref{prop:abelianizationfreeoocat} that $\sH^{\pol}(C)$ is given by the chain complex (seen as an object of $\ho(\Ch)$)

Let us show that $(\overline{\sH^{\pol}},\alpha^{\pol})$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with Thomason equivalences. Let $G$ and $\beta$ be as in the following $2$\nbd{}diagram

Let us show that $(\overline{\sH^{\pol}},\alpha^{\pol})$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with the Thomason equivalences. Let $G$ and $\beta$ be as in the following $2$\nbd{}diagram

Since $\sH^{\pol}$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with folk weak equivalences, there exists a unique $\delta : G \circ J \Rightarrow\sH^{\pol}$ that factorizes $\beta$ as

Since $\sH^{\pol}$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with the folk weak equivalences, there exists a unique $\delta : G \circ\J\Rightarrow\sH^{\pol}$ that factorizes $\beta$ as

@@ -709,7 +707,7 @@ Another consequence of the above counter-example is the following result, which

\ar[from=A,to=B,Rightarrow,"\delta'"]

\end{tikzcd}

\]

The uniqueness of such a factorization follows from a similar argument which is left to the reader. This proves that $\overline{\sH^{\pol}}$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with Thomason equivalences and in particular we have

The uniqueness of such a factorization follows from a similar argument which is left to the reader. This proves that $\overline{\sH^{\pol}}$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with the Thomason equivalences and in particular we have

\[

\sH^{\sing}\simeq\overline{\sH^{\pol}}.

\]

...

...

@@ -806,12 +804,12 @@ The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd{}cate

\]

is commutative.

It follows from what we said in \ref{paragr:compweakeq} that the morphism $\J$ cannot be an equivalence of op-prederivators. As we shall see later, $\J$ is not even homotopy cocontinuous. In particular, this implies that given a diagram $d : I \to\oo\Cat$, the canonical arrow of $\ho(\oo\Cat^{\Th})$

is an isomorphism on objects for every small category $I$, the above morphism simply reads

is the identity on objects for every small category $I$, morphism \eqref{equation:Jhocolim} simply reads

\[

\hocolim_I^{\Th}(d)\to\hocolim_I^{\folk}(d).

\]

...

...

@@ -880,11 +878,11 @@ The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd{}cate

where:

\begin{itemize}[label=-]

\item the top and bottom horizontal arrows are induced by $\pi$,

\item the middle horizontal arrow is induced by $\pi$ and the canonical morphism \[\hocolim_{i \in}^{\Th}(d_i)\to\hocolim_{i \in I}^{\folk}(d_i)\] from \ref{paragr:prelimcriteriongoodcat},

\item the middle horizontal arrow is induced by $\pi$ and the canonical morphism \[\hocolim_{i \inI}^{\Th}(d_i)\to\hocolim_{i \in I}^{\folk}(d_i)\] from \ref{paragr:prelimcriteriongoodcat},

\item the top vertical arrows are the canonical morphisms induced by every morphism of op-prederivators (see \ref{paragr:canmorphismcolimit}),

\item the bottom vertical arrows are induced by the co-cone \[(\varphi_i : d(i)\to\oo\Cat)_{i \in\Ob(I)}.\]

\end{itemize}

Since $\sH^{\pol}$ and $\sH^{\sing}$ are both homotopy cocontinuous (Proposition \ref{prop:singhmlgycocontinuous} and Proposition \ref{prop:polhmlgycocontinuous}), both top vertical arrows are isomorphisms. Hypotheses $(ii)$ and $(iii)$ imply that the bottom vertical arrows are isomorphisms and hypothesis $(i)$ imply that the top horizontal arrow is an isomorphism. By a 2-out-of-3 property, the bottom horizontal arrow is an isomorphism, which means exactly that $C$ is \good{}.

Since $\sH^{\pol}$ and $\sH^{\sing}$ are both homotopy cocontinuous (Proposition \ref{prop:singhmlgycocontinuous} and Proposition \ref{prop:polhmlgycocontinuous} respectively), both top vertical arrows are isomorphisms. Hypotheses $(ii)$ and $(iii)$ imply that the bottom vertical arrows are isomorphisms and hypothesis $(i)$ imply that the top horizontal arrow is an isomorphism. By a 2-out-of-3 property, the bottom horizontal arrow is an isomorphism, which means exactly that $C$ is \good{}.

\end{proof}

The previous proposition admits the following corollary, which will be of great use in later chapters.

\begin{corollary}\label{cor:usefulcriterion}

...

...

@@ -893,6 +891,7 @@ The previous proposition admits the following corollary, which will be of great

\begin{tikzcd}

A \ar[r,"u"]\ar[d,"f"]& B \ar[d,"g"]\\

C \ar[r,"v"]& D

\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]

\end{tikzcd}

\]

be a cocartesian square of $\oo\Cat$ such that:

...

...

@@ -918,7 +917,7 @@ The previous proposition admits the following corollary, which will be of great

\[

x \sim y \text{ when there exists } z : x \to y \text{ in } C_{n+1}.

\]

As it happens, we can use the adjunction $\tau_{\leq n}^{i}\dashv\iota_n$ to transport the folk model structure on$\oo\Cat$ to $n\Cat$.% Say that a morphism $f : C \to D$ of $n\Cat$ is a \emph{folk weak equivalence of $n$\nbd{}categories} (resp. \emph{folk trivial fibration of $n$\nbd{}categories}) if $\iota_n(f)$ is a weak equivalence (resp.\ fibration) for the folk model structure on $\oo\Cat$.

As it happens, we can use the adjunction $\tau_{\leq n}^{i}\dashv\iota_n$ to transport the folk model structure from$\oo\Cat$ to $n\Cat$.% Say that a morphism $f : C \to D$ of $n\Cat$ is a \emph{folk weak equivalence of $n$\nbd{}categories} (resp. \emph{folk trivial fibration of $n$\nbd{}categories}) if $\iota_n(f)$ is a weak equivalence (resp.\ fibration) for the folk model structure on $\oo\Cat$.

\end{paragr}

\begin{proposition}\label{prop:fmsncat}

There exists a model structure on $n\Cat$ such that:

...

...

@@ -940,12 +939,12 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

the functor $\tau^{i}_{\leq n}$ also preserves weak equivalences.

The functor $\tau^{i}_{\leq n} : \oo\Cat\to n\Cat$ sends weak equivalences of the folk model structure on $\oo\Cat$ to weak equivalences of the folk model structure on $n\Cat$.

The functor $\tau^{i}_{\leq n} : \oo\Cat\to n\Cat$ sends the weak equivalences of the folk model structure on $\oo\Cat$ to weak equivalences of the folk model structure on $n\Cat$.

\end{proposition}

\begin{proof}

Since every $\oo$\nbd{}category is fibrant for the folk model structure on

$\oo\Cat$\cite[Proposition 9]{lafont2010folk}, it suffices to show that

$\tau^{i}_{\leq n}$ sends folk trivial fibrations of $\oo\Cat$ to weak

$\tau^{i}_{\leq n}$ sends the folk trivial fibrations of $\oo\Cat$ to weak

equivalences of $n\Cat$ (in virtue of

Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model}).

% Let $C$ be an $\oo$\nbd{}category and let $\eta_C : C \to \iota_n\tau^{i}_{\leq

...

...

@@ -994,17 +993,17 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

$(k+1)$\nbd{}cell $\beta : f(x)\to f(y)$ 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 $k$\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

of $D$, $\beta ' : f(x)\to f(y)$, such that

\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

$f(\alpha')=\beta'$. 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 $l>n$

\item[Case $k=n$:] Since all $l$\nbd{}cells of $T(C)$ and $T(D)$ with $l>n$

are units, we trivially have that $f(x)=f(y)$ and $\beta$ is the unit on

$f(x)$. Now let $x'$ and $y'$ be parallel $n$\nbd{}cells of $C$ such that

$\eta_C(x')=x$ and $\eta_C(y')=y$ (this is always possible by definition

...

...

@@ -1020,7 +1019,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

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 $k>n$

\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 $x=y$.

\end{description}

Altogether, this proves that $T(f)$ is a folk trivial fibration, hence a folk

...

...

@@ -1033,7 +1032,7 @@ 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 $n$\nbd{}category $C'$ such that $\tau^{i}_{\leq n}(C')=C$.

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=-]

...

...

@@ -1044,7 +1043,7 @@ let $C'$ be the free $\oo$\nbd{}category such that:

\{(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 $k > n$ (i.e.\ $C'$ is an $(n+1)$\nbd{}category).

\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

...

...

@@ -1074,7 +1073,7 @@ We now turn to truncations of chain complexes.

Likewise $n$\nbd{}categories again, we can use the adjunction

@@ -1135,7 +1134,7 @@ We now turn to truncations of chain complexes.

\end{proof}

As a consequence of this lemma, we have the analogous of Proposition \ref{prop:truncationhomotopical}.

\begin{proposition}

The functor $\tau^{i}_{\leq n} : \Ch\to\Ch^{\leq n}$ sends weak equivalences of the projective model structure on $\Ch$ to weak equivalences of the projective model structure on $\Ch^{\leq n}$.

The functor $\tau^{i}_{\leq n} : \Ch\to\Ch^{\leq n}$ sends the weak equivalences of the projective model structure on $\Ch$ to weak equivalences of the projective model structure on $\Ch^{\leq n}$.

\end{proposition}

\begin{proof}

Let $f : K \to K'$ be a weak equivalence for the projective model structure on $\Ch$ and consider the naturality square

...

...

@@ -1155,14 +1154,14 @@ We now investigate the relation between truncation and abelianization.

\begin{paragr}

Let $C$ be $n$\nbd{}category. A straightforward computation shows that the chain complex $\lambda(\iota_n(C))$ is such that

\[

\lambda(\iota_n(C))_k=0

\lambda_k(\iota_n(C))=0

\]

for every $k > n$. Hence,$\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$ and we can define a functor $\lambda_{\leq n } : n\Cat\to\Ch^{\leq n}$ as

for every $k > n$ and thus$\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$. Hence, we can define a functor $\lambda_{\leq n } : n\Cat\to\Ch^{\leq n}$ as

@@ -1263,7 +1262,7 @@ As an immediate consequence of the previous lemma, the functor $\lambda_{\leq n}

is commutative (up to a canonical isomorphism).

\end{proposition}

\begin{proof}

Straightforward consequence of Lemma \ref{lemma:abelianizationtruncation} and the fact that the left derived functor of a composition of left Quillen functors is the composition of the left derived functors (see for example \cite[Theorem 1.3.7]{hovey2007model}.

Straightforward consequence of Lemma \ref{lemma:abelianizationtruncation} and the fact that the left derived functor of a composition of left Quillen functors is the composition of the left derived functors (see for example \cite[Theorem 1.3.7]{hovey2007model}).

\end{proof}

\begin{remark}

Beware that the square

...

...

@@ -1273,7 +1272,7 @@ As an immediate consequence of the previous lemma, the functor $\lambda_{\leq n}

\ho(\oo\Cat^{\folk})\ar[r,"\LL\lambda"]&\ho(\Ch)

\end{tikzcd}

\]

is \emph{not} commutative. If it were, then for every $n$\nbd{}category $C$ and every $k >n$, we would have $H_k^{\pol}(\iota_n(C))=0$ for every $k >n$, which is not even true for the case $n=1$ in the following chapter.

is \emph{not} commutative. If it were, then for every $n$\nbd{}category $C$ and every $k >n$, we would have $H_k^{\pol}(\iota_n(C))=0$ for every $k >n$, which is not even true for the case $n=1$as we shall see in the following chapter.

\end{remark}

A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the following corollary.

\begin{corollary}\label{cor:polhmlgycofibrant}

...

...

@@ -1358,9 +1357,10 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t

Now let $\Delta_{\leq2}$ be the full subcategory of $\Delta$ spanned by $[0]$, $[1]$ and $[2]$ and $i : \Delta_{\leq2}\to\Delta$ the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta}\to\Psh{\Delta_{\leq2}}$ which has a right-adjoint $i_* : \Psh{\Delta}\to\Psh{\Delta_{\leq2}}$. 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

Using the description of $\Or_0$, $\Or_1$ and $\Or_2$ from \ref{paragr:orientals}, we deduce that a morphism $F : i^*(N_{\oo}(C))\to i^*(N_1(D))$ of $\Psh{\Delta_{\leq2}}$ consists of a function $F_0 : C_0\to D_0$ and a function $F_1 : C_1\to D_1$ such that

\begin{itemize}[label=-]

\item for every $x \in C_0$, we have $F_1(1_x)=1_{F_0(x)}$,

...

...

@@ -1378,11 +1378,11 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t

\end{itemize}

In particular, it follows that $F_1$ is compatible with composition of $1$\nbd{}cells in an obvious sense and that for every $2$\nbd{}cell $\alpha : f \Rightarrow g$ of $C$, we have $F_1(f)=F_1(g)$. This means exactly that we have a natural isomorphism

@@ -1458,9 +1458,9 @@ Finally, we obtain the result we were aiming for.

\]

However, it is still an open question to know whether for $k \in\{2,3\}$ we have

\[

H^{\sing}_k(C)\simeq H^{\pol}_k(C).

H^{\sing}_k(C)\simeq H^{\pol}_k(C)

\]

The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of $k$ is the analogue of Lemma \ref{lemma:truncationcounit}. But contrary to the case $k=1$, it is not generally true that the canonical morphism $c_k N_{\oo}(C)\to\tau^{i}_{\leq k}(C)$ is an isomorphism when $k \geq2$. However, what we really need is that the image by $\lambda$ of this morphism be a quasi-isomorphism. In the case $k=2$, it seems that this canonical morphism admits an oplax $2$\nbd{}functor as an inverse up to oplax transformation which could be an hint towards the conjecture that $H^{\sing}_2(C)\simeq H^{\pol}_2(C)$ for every $\oo$\nbd{}category $C$.

for every $\oo$\nbd{}category $C$. The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of $k$ is the analogue of Lemma \ref{lemma:truncationcounit}. But contrary to the case $k=1$, it is not generally true that the canonical morphism $c_k N_{\oo}(C)\to\tau^{i}_{\leq k}(C)$ is an isomorphism when $k \geq2$. However, what we really need is that the image by $\lambda$ of this morphism be a quasi-isomorphism. In the case $k=2$, it seems that this canonical morphism admits an oplax $2$\nbd{}functor as an inverse up to oplax transformation which could be an hint towards the conjecture that $H^{\sing}_2(C)\simeq H^{\pol}_2(C)$ for every $\oo$\nbd{}category $C$.

@@ -219,7 +219,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\begin{remark}

Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$ by Ara and Maltsiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.

\end{remark}

By definition, for all $1\leq n \leq m \leq\omega$, the canonical inclusion \[n\Cat\hookrightarrow m\Cat\] sends Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivator $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.

By definition, for all $1\leq n \leq m \leq\omega$, the canonical inclusion \[n\Cat\hookrightarrow m\Cat\] sends the Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivator $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.

\begin{proposition}

For all $1\leq n \leq m \leq\omega$, the canonical morphism

\[

...

...

@@ -605,12 +605,12 @@ For later reference, we put here the following trivial but important lemma, whos

\fi

\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason equivalences}

\begin{lemma}\label{lemma:nervehomotopical}

The nerve functor $N_{\omega} : \omega\Cat\to\Psh{\Delta}$ sends equivalences of $\omega$-categories to weak equivalences of simplicial sets.

The nerve functor $N_{\omega} : \omega\Cat\to\Psh{\Delta}$ sends the equivalences of $\omega$-categories to weak equivalences of simplicial sets.

\end{lemma}

\begin{proof}

Since every $\omega$-category is fibrant for the folk model structure \cite[Proposition 9]{lafont2010folk}, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that it suffices to show that the nerve sends folk trivial fibrations to weak equivalences of simplicial sets. In particular, it suffices to show the stronger condition that the nerve sends folk trivial fibrations to trivial fibrations of simplicial sets.

Since every $\omega$-category is fibrant for the folk model structure \cite[Proposition 9]{lafont2010folk}, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that it suffices to show that the nerve sends folk the trivial fibrations to weak equivalences of simplicial sets. In particular, it suffices to show the stronger condition that the nerve sends the folk trivial fibrations to trivial fibrations of simplicial sets.

By adjunction, this is equivalent to showing that the functor $c_{\omega} : \Psh{\Delta}\to\omega\Cat$ sends cofibrations of simplicial sets to folk cofibrations. Since $c_{\omega}$ is cocontinuous and the cofibrations of simplicial sets are generated by the inclusions

By adjunction, this is equivalent to showing that the functor $c_{\omega} : \Psh{\Delta}\to\omega\Cat$ sends the cofibrations of simplicial sets to folk cofibrations. Since $c_{\omega}$ is cocontinuous and the cofibrations of simplicial sets are generated by the inclusions