It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left vertical morphisms of the above square are isomorphisms. Then, an immediate computation left to the reader shows that $\sD_0$ is \good{} and that $\sH^{\pol}(\sD_0)\simeq\sH^{\sing}(\sD_0)\simeq\mathbb{Z}$. By a 2-out-of-3 property, we deduce that $\pi_C : \sH^{\sing}(C)\to\sH^{\pol}(C)$ is an isomorphism.
It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and
Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left vertical
morphisms of the above square are isomorphisms. Then, an immediate computation
left to the reader shows that $\sD_0$ is \good{} and that
$\sH^{\pol}(\sD_0)\simeq\sH^{\sing}(\sD_0)\simeq\mathbb{Z}$. By a 2-out-of-3
property, we deduce that $\pi_C : \sH^{\sing}(C)\to\sH^{\pol}(C)$ is an
isomorphism and $\sH^{\pol}(C)\simeq\sH^{\sing}(C)\simeq\mathbb{Z}$.
\end{proof}
\begin{remark}
Definition \ref{def:contractible} admits an obvious ``lax'' variation and Proposition \ref{prop:contractibleisgood} is also true for lax contractible $\oo$\nbd{}categories.
\end{remark}
We end this section with an important result on slices$\oo$\nbd{}category (Paragraph \ref{paragr:slices}).
We end this section with an important result on slice $\oo$\nbd{}categories (Paragraph \ref{paragr:slices}).
\begin{proposition}\label{prop:slicecontractible}
Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. The $\oo$\nbd{}category $A/a_0$ is oplax contractible.
\end{proposition}
...
...
@@ -84,7 +90,7 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para
\[
r(p(x))=\1^k_{\trgt_0(e_n)},
\]
where we wrote $p$ for the unique $\oo$\nbd{}functor $\sD_n \to\sD_0$.
where we write $p$ for the unique $\oo$\nbd{}functor $\sD_n \to\sD_0$.
Now for $0\leq k <n$, we define $\alpha_{\src_k(e_n)}$ and $\alpha_{\trgt_k(e_n)}$ as
\[
...
...
@@ -98,7 +104,6 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para
In particular, for every $n \in\mathbb{N}$, $\sD_n$ is \good{}. Recall from \ref{paragr:inclusionsphereglobe} that for every $n \geq0$, we have a cocartesian square
\[
...
...
@@ -119,7 +124,7 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re
is cocartesian.
\end{lemma}
\begin{proof}
Since colimits in presheaves categories are computed pointwise, what we need
Since colimits in presheaf categories are computed pointwise, what we need
to show is that for every $k\geq0$, the following commutative square is
cocartesian
\begin{equation}\label{squarenervesphere}
...
...
@@ -158,7 +163,8 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re
Now, let $\varphi : \Or_k \to\sS_n$ be an $\oo$\nbd{}functor. There are several cases to distinguish.
\begin{description}
\item[Case $k<n$:] Since every generating cell of $\gamma$ of $\Or_k$ is of dimension non-greater than $k$, the cell $\varphi(\gamma)$ is of dimension strictly lower than $n$. Since all cells of dimension strictly lower than $n$ are both in the image of $j^+_n$ and in the image of $j^-_n$, $\varphi$ obviously factors through $j^+_n$ (and $j^+_n$).
\item[Case $k<n$:] Since every generating cell of $\gamma$ of $\Or_k$ is of
dimension not greater than $k$, the cell $\varphi(\gamma)$ is of dimension strictly lower than $n$. Since all cells of dimension strictly lower than $n$ are both in the image of $j^+_n$ and in the image of $j^-_n$, $\varphi$ obviously factors through $j^+_n$ (and $j^-_n$).
\item[Case $k=n$:] The image of $\alpha_n$ is either a non-trivial $n$\nbd{}cell of $\sS_n$ or a unit on a strictly lower dimensional cell. In the second situation, everything works like the case $k<n$. Now suppose for example that $\varphi(\alpha_n)$ is $h^+_n$, which is in the image of $j^+_n$. Since all of the other generating cells of $\Or_n$ are of dimension strictly lower than $n$, their images by $\varphi$ are also of dimension strictly lower than $n$ and hence, are all contained in the image of $j^+_n$. Altogether this proves that $\varphi$ factors through $j^+_n$. The case where $\varphi(\alpha_n)=h^-_n$ is symmetric.
\item[Case $k>n$:] Since $\sS_n$ is an $n$\nbd{}category, the image of
$\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$
...
...
@@ -248,7 +254,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
is Thomason homotopy cocartesian, then $C$ is \good{}.
\end{proposition}
\begin{proof}
Since the morphisms $i_k$ are folk cofibration and the
Since the morphisms $i_k$ are folk cofibrations and the
$\oo$\nbd{}categories $\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant
and \good{}, it follows from Corollary \ref{cor:usefulcriterion} and
an immediate induction that all $\sk_k(C)$ are \good{}. The result
...
...
@@ -289,7 +295,7 @@ higher than $1$.
(a',p)&\mapsto a'.
\end{aligned}
\]
This is a special case of the more general notion of slice $\oo$\nbd{}categories introduced in \ref{paragr:slices}. In particular, given an $\oo$\nbd{}category $X$ and an $\oo$\nbd{}functor $f : X \to A$, we have defined the $\oo$\nbd{}category $X/a$ and the $\oo$\nbd{}functor
This is a special case of the more general notion of slice $\oo$\nbd{}category introduced in \ref{paragr:slices}. In particular, given an $\oo$\nbd{}category $X$ and an $\oo$\nbd{}functor $f : X \to A$, we have defined the $\oo$\nbd{}category $X/a$ and the $\oo$\nbd{}functor
\[
f/a : X/a \to A/a
\]
...
...
@@ -303,7 +309,7 @@ higher than $1$.
More explicitly, the $n$\nbd{}cells of $X/a$ can be described as pairs $(x,p)$
where $x$ is an $n$\nbd{}cell of $X$ and $p$ is an arrow of $A$ of the form
\[
p : f(x)\top\text{ if }n=0
p : f(x)\toa\text{ if }n=0
\]
and
\[
...
...
@@ -320,7 +326,7 @@ higher than $1$.
\[
(x,p)\mapsto(f(x),p),
\]
and the canonical $\oo$\nbd{}functor $X \to X/a$ as
and the canonical $\oo$\nbd{}functor $X/a\to X$ as
\[
(x,p)\mapsto x.
\]
...
...
@@ -331,7 +337,7 @@ higher than $1$.
X/\beta : X/a &\to X/{a'}\\
(x,p) &\mapsto (x,\beta\circ p),
\end{align*}
which takes part of a commutative triangle
which takes part in a commutative triangle
\[
\begin{tikzcd}[column sep=tiny]
X/{a}\ar[rr,"X/{\beta}"]\ar[dr]&& X/{a'}\ar[dl]\\
...
...
@@ -442,7 +448,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
\end{lemma}
\begin{proof}
It is immediate to check that for every object $a$ of $A$, the canonical
forgetful functor $\pi_{a} : A/a \to A$ is a Conduché functor (see Section
forgetful functor $\pi_{a} : A/a \to A$ is a discrete Conduché functor (see Section
\ref{section:conduche}). Hence, from Lemma \ref{lemma:pullbackconduche} we
know that $X/a \to X$ is a discrete Conduché $\oo$\nbd{}functor. The result follows
then from Theorem \ref{thm:conduche}.
...
...
@@ -473,7 +479,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
\item[(OR1)] Each $\Or_n$ is a free $\omega$-category whose set of generating $k$-cells is canonically isomorphic the sets of increasing sequences
\item[(OR1)] Each $\Or_n$ is a free $\oo$\nbd{}category whose set of generating
$k$\nbd{}cells is canonically isomorphic to the sets of increasing sequences
\[
0\leq i_1 < i_2 < \cdots < i_k \leq n,
\]
...
...
@@ -33,7 +34,7 @@
\end{description}
We use the notation $\langle i_1\, i_2\cdots i_k\rangle$ for such a cell. In particular, we have that:
\begin{itemize}[label=-]
\item There is no generating $k$-cells for $k>n$. Hence, $\Or_n$ is an $n$\nbd{}category.
\item There are no generating $k$-cells for $k>n$. Hence, $\Or_n$ is an $n$\nbd{}category.
\item There is exactly one generating $n$\nbd{}cell of $\Or_n$, which is $\langle0\,1\cdots n\rangle$. We refer to this cell as the \emph{principal cell of $\Or_n$}.
\item There are exactly $n+1$ generating $(n-1)$-cells of $\Or_n$. They correspond to the maps
\[
...
...
@@ -42,7 +43,7 @@
for $i \in\{0,\cdots,n\}$.
\end{itemize}
\begin{description}
\item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composition of all the generating $(n-1)$\nbd{}cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd{}cell appearing exactly once in the composite.
\item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composition of all the generating $(n-1)$\nbd{}cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd{}cells appearing exactly once in the composite.
\end{description}
Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight (see \ref{paragr:weight}) of the $(n-1)$\nbd{}cells corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$.
Here are some pictures in low dimension:
...
...
@@ -179,7 +180,12 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
for the localization morphism. The reason is to avoid confusion with other weak equivalences on $n\Cat$ that we will introduce later.
\end{paragr}
\begin{paragr}
By definition, the nerve functor induces a morphism of localizers ${N_n : (n\Cat,\W_n^{\Th})\to(\Psh{\Delta},\W_{\Delta})}$ and hence a morphism of op\nbd{}prederivators
By definition, the nerve functor induces a morphism of localizers
@@ -188,7 +194,8 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
For every $1\leq n \leq\oo$, the morphism \[{\overline{N}_n : \Ho(n\Cat^\Th)\to\Ho(\Psh{\Delta})}\] is an equivalence of op\nbd{}prederivators.
\end{theorem}
\begin{proof}
In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta}\to\Psh{\Delta}$, as well as a zigzag of morphisms of functors
Recall from \ref{paragr:nerve} that $c_n : \Psh{\Delta}\to n\Cat$ denotes
the left adjoint to the nerve functor $N_n$. In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta}\to\Psh{\Delta}$, as well as a zigzag of morphisms of functors
@@ -210,7 +217,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
For every $1\leq n \leq\oo$, the localizer $(n\Cat^{\Th},\W_n^{\Th})$ is homotopy cocomplete (Definition \ref{def:cocompletelocalizer}).
\end{corollary}
We will speak of ``Thomason homotopy colimits'' and ``Thomason homotopy
cocartesian square'' for homotopy colimits and homotopy cocartesian squares in
cocartesian squares'' for homotopy colimits and homotopy cocartesian squares in
the localizer $(n\Cat^{\Th},\W_n^{\Th})$.
Another consequence of Gagna's theorem is the following
...
...
@@ -222,9 +229,9 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
This follows immediately from the fact that $\overline{N_n} : \ho(n\Cat^{\Th})\to\ho(\Psh{\Delta})$ is an equivalence of categories and the fact that weak equivalences of simplicial sets are saturated (because they are the weak equivalences of a model structure).
\end{proof}
\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.
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{}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{}prederivators$\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.
\begin{proposition}
For all $1\leq n \leq m \leq\omega$, the canonical morphism
\[
...
...
@@ -334,7 +341,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\]
for every cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$\nbd{}functor
\[
\sD_1\otimes C \overset{p\otimesi}{\longrightarrow}\sD_0\otimes D \simeq D,
\sD_1\otimes C \overset{p\otimesu}{\longrightarrow}\sD_0\otimes D \simeq D,
\]
where $p$ is the only $\oo$\nbd{}functor $\sD_1\to\sD_0$.
\end{paragr}
...
...
@@ -394,7 +401,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{enumerate}
Furthermore, $i$ is a \emph{strong oplax deformation retract} if $\alpha$ can be chosen such that:
\begin{enumerate}[label=(\alph*),resume]
\item$\alpha\ast i =1_i$.
\item$\alpha\star i =1_i$.
\end{enumerate}
An oplax deformation retract is a particular case of homotopy equivalence and thus of Thomason equivalence.
\end{paragr}
...
...
@@ -434,7 +441,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
we deduce the existence of $r' : B' \to A'$ that makes the whole diagram commutes. In particular, we have $r' \circ i' =\mathrm{id}_{A'}$.
we deduce the existence of $r' : B' \to A'$ that makes the whole diagram commute. In particular, we have $r' \circ i' =\mathrm{id}_{A'}$.
From the commutativity of (\ref{diagramstrong}), we easily deduce the commutativity of the following solid arrow diagram
\[
...
...
@@ -558,7 +565,10 @@ For later reference, we put here the following trivial but important lemma, whos
The model structure of the previous theorem is commonly referred to as \emph{folk model structure} on $\omega\Cat$.
Data of this model structure will often be referred to by using the adjective folk, e.g.\ \emph{folk cofibration}. Consequently \emph{folk weak equivalence} and \emph{equivalence of $\oo$\nbd{}categories} mean the same thing.
Furthermore, as in the Thomason case (see \ref{paragr:notationthom}), we usually make reference to the word ``folk'' in the notations of homotopic constructions induced by the folk weak equivalences. For example, we write $\W^{\folk}$ the class of folk weak equivalences, $\Ho(\oo\Cat^{\folk})$ for the homotopy op\nbd{}prederivator of $(\oo\Cat,\W_{\oo}^{\folk})$ and
Furthermore, as in the Thomason case (see \ref{paragr:notationthom}), we
usually make reference to the word ``folk'' in the notations of homotopic
constructions induced by the folk weak equivalences. For example, we write
$\W^{\folk}$ for the class of folk weak equivalences, $\Ho(\oo\Cat^{\folk})$ for the homotopy op\nbd{}prederivator of $(\oo\Cat,\W_{\oo}^{\folk})$ and
\[
\gamma^{\folk} : \oo\Cat\to\Ho(\oo\Cat^{\folk})
\]
...
...
@@ -613,10 +623,13 @@ 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 the equivalences of $\omega$-categories to weak equivalences of simplicial sets.
The nerve functor $N_{\omega} : \omega\Cat\to\Psh{\Delta}$ sends the equivalences of $\oo$\nbd{}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 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.
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 the folk 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 the cofibrations of simplicial sets to folk cofibrations. Since $c_{\omega}$ is cocontinuous and the cofibrations of simplicial sets are generated by the inclusions
\[
...
...
@@ -792,7 +805,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
&(x_n,b_{n+1})
\end{pmatrix}
\]
where the $x_i$ and $x'_i$ are $i$-cells of $A$, the $b_i$ and $b'_i$ are $i$-cells of $B$, such that
where the $x_i$ and $x'_i$ are $i$-cells of $A$, and the $b_i$ and $b'_i$ are $i$-cells of $B$, such that
\[
\begin{pmatrix}
\begin{matrix}
...
...
@@ -844,7 +857,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
\begin{proof}
Before anything else, recall from Lemma \ref{lemma:ooequivalenceisfunctorial} that given an $\oo$\nbd{}functor $F : X \to Y$ and $n$\nbd{}cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x)\sim_{\oo} F(y)$.
\begin{enumerate}[label=(\roman*)]
\item Let $b_0$ be $0$\nbd{}cell of $B$ and set $c_0:=v(b_0)$. By definition, the pair $(b_0,1_{c_0})$ is a $0$-cell of $B/c_0$. By hypothesis, we know that there exists a $0$\nbd{}cell $(a_0,c_1)$ of $A/c_0$ such that $(u(a_0),c_1)\sim_{\oo}(b_0,1_{c_0})$. Hence, by applying the canonical functor $B/c_0\to B$, we obtain that $u(a_0)\sim_{\oo} b_0$.
\item Let $b_0$ be a $0$\nbd{}cell of $B$ and set $c_0:=w(b_0)$. By definition, the pair $(b_0,1_{c_0})$ is a $0$-cell of $B/c_0$. By hypothesis, we know that there exists a $0$\nbd{}cell $(a_0,c_1)$ of $A/c_0$ such that $(u(a_0),c_1)\sim_{\oo}(b_0,1_{c_0})$. Hence, by applying the canonical functor $B/c_0\to B$, we obtain that $u(a_0)\sim_{\oo} b_0$.
\item Let $f$ and $f'$ be parallel $n$\nbd{}cells of $A$ and $\beta : u(f)\to u(f')$ an $(n+1)$\nbd{}cell of $B$. We need to show that there exists an $(n+1)$\nbd{}cell $\alpha : f \to f'$ of $A$ such that $u(\alpha)\sim_{\oo}\beta$.
...
...
@@ -894,7 +907,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen
&(\alpha,\Lambda)
\end{pmatrix}
\]
whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha\sim_{\oo}\beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0\to A$, .
whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha\sim_{\oo}\beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0\to A$.
