Note that for $n=0$, the above formula reads $S_0(C)=C_0$. The face

operators $\partial_i : S_{n}(C)\to S_{n-1}(C)$ are induced by horizontal

composition and the degeneracy operators $s_i : S_{n}(C)\to S_{n+1}(C)$ are

Note that for $n=0$, the above formula reads $S_0(C)=C_0$. For $n>0$, the face operators $\partial_i : S_{n}(C)\to S_{n-1}(C)$ are induced by horizontal

composition for $0 < i <n$ and by projection for $i=0$ or $n$. The degeneracy operators $s_i \colon S_{n}(C)\to S_{n+1}(C)$ are

induced by the units for the horizontal composition.

Post-composing $S(C)$ with the nerve functor $N : \Cat\to\Psh{\Delta}$, we

Let $A$ be a $1$\nbd{}category and $a$ an object of $A$. Recall that we write $A/a$ for the slice $1$\nbd{}category of $A$ over $a$, that is the $1$\nbd{}category whose description is as follows:

\begin{itemize}[label=-]

\item an object of $A/a$ is a pair $(a', p : a' \to a)$ where $a'$ is an object of $A$ and $p$ is an arrow of $A$,

\item an arrow $(a',p)\to(a'',p')$ of $A/a$ is an arrow $ q : a' \to a''$ of $A$ such that $p'\circ q = p$,

\item an arrow of $A/a$ is a pair $(q,p : a' \to a)$ where $p$ is an arrow of

$A$ and $q$ is an arrow of $A$ of the form $q : a'' \to a'$. The target of

$(q,p)$ is given by $(a',p)$ and the source by $(a'',p\circ q)$. % $ q : a' \to a''$ of $A$ such that $p'\circ q = p$.

\end{itemize}

and we write $\pi_a$ for the canonical forgetful functor

We write $\pi_a$ for the canonical forgetful functor

\[

\begin{aligned}

\pi_{a} : A/a &\to A \\

...

...

@@ -433,7 +435,9 @@ higher than $1$.

we obtain that every $1$\nbd{}category $A$ is (canonically isomorphic to) the colimit

\[

\colim_{a \in A}(A/a).

\]

\]

% In other words, this simply say that the colimit of the Yoneda embedding $A \to

% \Psh{A}$ is the terminal presheaves

We now proceed to prove that this colimit is homotopic with respect to

the folk weak equivalences.

\end{paragr}

...

...

@@ -532,24 +536,32 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func

Beware that in the previous corollary, we did \emph{not} suppose that $X$ was free.

\begin{proof}

Let $P$ be a free $\omega$-category and $g : P \to X$ a folk trivial fibration

and consider the following commutative square of $\ho(\oo\Cat^{\folk})$

and consider the following commutative diagram of $\ho(\oo\Cat^{\folk})$

\displaystyle\hocolim^{\folk}_{a \in A}(P/a) \ar[d]\ar[r]&\displaystyle\colim_{a \in A}(P/a) \ar[d]\ar[r]& P \ar[d]\\

\displaystyle\hocolim^{\folk}_{a \in A}(X/a) \ar[r]&\displaystyle\colim_{a \in A}(X/a) \ar[r]& X

\end{tikzcd}

\end{equation}

where the vertical arrows are induced by the arrows

where the middle and most left vertical arrows are induced by the arrows

\[

g/a : P/a \to X/a.

g/a : P/a \to X/a,

\]

Since trivial fibrations are stable by pullback, $g/a$ is a trivial fibration. This proves that the left vertical arrow of square (\ref{comsquare}) is an isomorphism.

and the most right vertical arrow is induced by $g$.

Since trivial fibrations are stable by pullback, $g/a$ is a trivial fibration.

This proves that the most left vertical arrow of diagram \eqref{comsquare} is an isomorphism.

Moreover, from Paragraph \ref{paragr:unfolding} and Lemma \ref{lemma:colimslice}, we deduce that the right vertical arrow of (\ref{comsquare}) can be identified with the image of $g : P \to X$ in $\ho(\omega\Cat)$ and hence is an isomorphism.

Finally, from Proposition \ref{prop:sliceiscofibrant} and Corollary \ref{cor:cofprojms}, we deduce that the top horizontal arrow of (\ref{comsquare}) is an isomorphism.

From Proposition \ref{prop:sliceiscofibrant} and Corollary

\ref{cor:cofprojms}, we deduce that the arrow \[\hocolim_{a \in

A}^{\folk}(P/a)\to\colim_{a \in A}(P/a)\] is an isomorphism. Moreover, from Lemma \ref{lemma:colimslice}, we know that the arrows

\[\colim_{a \in A}(P/a)\to P\] and \[\colim_{a \in A}(X/a)\to X\] are

isomorphisms.

By an immediate 2-out-of-3 property, this proves the result.

Finally, since $g$ is a folk weak equivalence, the most right vertical arrow of diagram \eqref{comsquare} is an

isomorphism and by an immediate 2-out-of-3 property this proves

that all arrows of \eqref{comsquare} are isomorphisms. In particular, so is

the composition of the two bottom horizontal arrows, which is what we desired

to show.

\end{proof}

We now move on to the next step needed to prove that every $1$\nbd{}category is \good{}. For that purpose, let us recall a construction commonly referred to as the ``Grothendieck construction''.

@@ -218,7 +218,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\end{corollary}

We will speak of ``Thomason homotopy colimits'' and ``Thomason homotopy

cocartesian squares'' for homotopy colimits and homotopy cocartesian squares in

the localizer $(n\Cat^{\Th},\W_n^{\Th})$.

the localizer $(n\Cat^{\Th},\W_n^{\Th})$. (See also \ref{paragr:thomhmtpycol} below.)

Another consequence of Gagna's theorem is the following

corollary.

...

...

@@ -232,7 +232,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

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 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{}prederivators $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.

\begin{proposition}

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

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

\[

\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})

...

...

@@ -248,6 +248,21 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\end{tikzcd}

\]

\end{proof}

\begin{paragr}\label{paragr:thomhmtpycol}

It follows from the previous proposition that for all $1\leq n \leq m \leq

\omega$, the morphism $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$ of

op\nbd{}prederivators is homotopy cocontinuous and reflects homotopy

colimits (in an obvious sense). Hence, given a diagram $d : I

\to n\Cat$ with $n>0$, we can harmlessly use the notation

\[

\hocolim^{\Th}_{i \in I}(d)

\]

for both the Thomason homotopy colimits in $n\Cat$ and in $\oo\Cat$ (or any

$m\Cat$ with $n\leq m$). Similarly, a commutative square of

$n\Cat$ is Thomason homotopy cocartesian in $n\Cat$ if and

only if it is so in $\oo\Cat$. Hence, there is really no ambiguity when simply

calling such a square \emph{Thomason homotopy cocartesian}.

\end{paragr}

\section{Tensor product and oplax transformations}

Recall that $\oo\Cat$ can be equipped with a monoidal product $\otimes$, introduced by Al-Agl and Steiner in \cite{al1993nerves} and by Crans in \cite{crans1995combinatorial}, commonly referred to as the \emph{Gray tensor product}. The implicit reference for this section is \cite[Appendices A and B]{ara2016joint}.

\begin{paragr}

...

...

@@ -291,7 +306,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\begin{tikzcd}

& Y \\

X \ar[ru,"u"]\ar[r,"\alpha"]\ar[rd,"v"']&\homlax(\sD_1,Y)\ar[u,"\pi_0^Y"']\ar[d,"\pi_1^Y"]\\

& Y

& Y,

\end{tikzcd}

\]

where $\pi^Y_0$ and $\pi^Y_1$ are induced by the two $\oo$\nbd{}functors $\sD_0\to\sD_1$ and where we implicitly used the isomorphism $\homlax(\sD_0,Y)\simeq Y$, is commutative.