Commit 39423129 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Edited a lot of typos and started to do the corrections suggested by Garner

parent 1b49db26
......@@ -59,7 +59,7 @@ In this section, we review some homotopical results on free
Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the
description of the arrows of $L(G)$ given in the previous paragraph shows that
$C$ is free and that its set of generating $1$-cells is (isomorphic to) the
non unital $1$-cells of $G$.
set of non unital $1$-cells of $G$.
\end{proof}
\begin{remark}
In other words, a category is free on a graph if and only if it is free on a
......@@ -85,10 +85,10 @@ In this section, we review some homotopical results on free
\]
For a graph $G$, the simplicial set $i_!(G)$ has $G_0$ as its set of
$0$-simplices, $G_1$ as its set of $1$-simplices and all $k$-simplices are
degenerated for $k>1$. For future reference, we put here the following lemma.
degenerate for $k>1$. For future reference, we put here the following lemma.
\end{paragr}
\begin{lemma}\label{lemma:monopreserved}
The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphism.
The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphisms.
\end{lemma}
\begin{proof}
What we need to show is that, given a morphism of simplicial sets
......@@ -96,7 +96,7 @@ In this section, we review some homotopical results on free
f : X \to Y,
\]
if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all
$n$-simplices of $X$ are degenerated for $n\geq 2$, then $f$ is a
$n$\nbd{}simplices of $X$ are degenerate for $n\geq 2$, then $f$ is a
monomorphism. A proof of this assertion is contained in \cite[Paragraph
3.4]{gabriel1967calculus}. The key argument is the Eilenberg--Zilber Lemma
(Proposition 3.1 of op. cit.).
......@@ -112,7 +112,7 @@ In this section, we review some homotopical results on free
\ar[r,"f_n"]& X_{n}
\end{tikzcd}
\]
of arrows of $C$. Such an $n$-simplex is degenerated if and only if at least
of arrows of $C$. Such an $n$-simplex is degenerate if and only if at least
one of the $f_k$ is a unit. It is straightforward to check that the composite
of
\[
......@@ -125,7 +125,7 @@ In this section, we review some homotopical results on free
\Cat.
\]
We now review a construction due Dwyer and Kan
We now review a construction due to Dwyer and Kan
(\cite{dwyer1980simplicial}). Let $G$ be a reflexive graph. For
every $k\geq 1$, we define the simplicial set $N^k(G)$ as the
sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains
......@@ -137,7 +137,7 @@ In this section, we review some homotopical results on free
\]
of arrows of $L(G)$ such that
\[
\sum_{1 \leq i \leq n}\ell(f_i) \leq n.
\sum_{1 \leq i \leq n}\ell(f_i) \leq k.
\] In particular, we have
\[
N^1(G)=i_!(G)
......@@ -238,7 +238,7 @@ From the previous proposition, we deduce the following very useful corollary.
L(C) \ar[r,"L(\gamma)"]& L(D)
\end{tikzcd}
\]
is a Thomason \emph{homotopy} cocartesian square of $\Cat$.
is a Thomason homotopy cocartesian square of $\Cat$.
\end{corollary}
\begin{proof}
Since the nerve $N$ induces an equivalence of op-prederivators
......@@ -315,7 +315,7 @@ From the previous proposition, we deduce the following very useful corollary.
$\beta$. For each element $x$ of $E$, we denote by $F_x$ the ``fiber'' of $x$,
that is the set of objects of $A$ that $\beta$ sends to $x$. We consider the
set $E$ and each $F_x$ as discrete reflexive graphs, i.e.\ reflexive graphs
with no non-unit arrows. Now, let $G$ be the reflexive graph defined with the
with no non-unit arrows. Now, let $G$ be the reflexive graph defined by the
following cocartesian square
\[
\begin{tikzcd}
......@@ -328,9 +328,9 @@ From the previous proposition, we deduce the following very useful corollary.
only one that sends an element $a \in F_x$ to $x$. In other words, $G$ is
obtained from $A$ by collapsing the objects that are identified through
$\beta$. It admits the following explicit description: $G_0$ is (isomorphic
to) $E$ and the set of non-units arrows of $G$ is (isomorphic to) the set of
non-units arrows of $A$; the source (resp. target) of a non-unit arrow $f$ of
$G$ is the source (resp. target) of $\beta(f)$. This completely describe $G$.
to) $E$ and the set of non-unit arrows of $G$ is (isomorphic to) the set of
non-unit arrows of $A$; the source (resp. target) of a non-unit arrow $f$ of
$G$ is the source (resp. target) of $\beta(f)$. This completely describes $G$.
% Notice also for later reference that the morphism \[ \coprod_{x \in E}F_x
% \to A\] is a monomorphism, i.e. injective on objects and arrows.
......@@ -343,12 +343,12 @@ From the previous proposition, we deduce the following very useful corollary.
\end{tikzcd}
\]
where the arrow $E \to B$ is the canonical inclusion. Hence, by universal
property, the dotted arrow exists and makes the whole diagram commutes. A
property, the dotted arrow exists and makes the whole diagram commute. A
thorough verification easily shows that the morphism $G \to B$ is a
monomorphism of $\Rgrph$.
By forming successive cocartesian square and combining with the square
obtained earlier, we obtain a diagram of three cocartesian square:
By forming successive cocartesian squares and combining with the square
obtained earlier, we obtain a diagram of three cocartesian squares:
\[
\begin{tikzcd}[row sep = large]
\displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]&\\
......@@ -383,7 +383,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref{prop:hmtpysquaregraphbetter} to a few examples.
\begin{example}[Identifying two objects]\label{example:identifyingobjects}
Let $C$ be a free category, $A$ and $B$ be two objects of $C$ with $A\neq B$ and let $C'$ be
the category obtained from $C$ by identifying $A$ and $B$, i.e.\ defined with
the category obtained from $C$ by identifying $A$ and $B$, i.e.\ defined by
the following cocartesian square
\[
\begin{tikzcd}
......@@ -399,7 +399,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\begin{example}[Adding a generator]
Let $C$ be a free category, $A$ and $B$ two objects of $C$ (possibly equal)
and let $C'$ be the category obtained from $C$ by adding a generator $A \to
B$, i.e.\ defined with the following cocartesian square:
B$, i.e.\ defined by the following cocartesian square:
\[
\begin{tikzcd}
\sS_0 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\
......@@ -422,7 +422,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\begin{example}[Identifying two generators]
Let $C$ be a free category and let $f,g : A \to B$ be parallel generating arrows of
$C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by
``identifying'' $f$ and $g$, i.e. defined with the following cocartesian
``identifying'' $f$ and $g$, i.e. defined by the following cocartesian
square
\[
\begin{tikzcd}
......@@ -443,9 +443,9 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref{cor:hmtpysquaregraph}.
\end{example}
\begin{example}[Killing a generator]\label{example:killinggenerator}
Let $C$ be a free category and let $f : A \to B$ be one of its generating arrow
Let $C$ be a free category and let $f : A \to B$ be one of its generating arrows
such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by
``killing'' $f$, i.e. defined with the following cocartesian square:
``killing'' $f$, i.e. defined by the following cocartesian square:
\[
\begin{tikzcd}
\sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\
......@@ -591,7 +591,6 @@ bisimplicial sets.
\[
f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})
\]
is a weak equivalence of simplicial sets. Recall now a very useful lemma.
\end{paragr}
\begin{lemma}\label{bisimpliciallemma}
......@@ -802,7 +801,7 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
for all $n,m \geq 0$.
\end{definition}
\begin{paragr}\label{paragr:formulabisimplicialnerve}
In other words, the bisimplicial nerve of $C$ is obtained by ``un-curryfying''
In other words, the bisimplicial nerve of $C$ is obtained by ``un-currying''
the functor $NS(C) : \Delta^{op} \to \Psh{\Delta}$.
Since the nerve $N$ commutes with products and sums, we obtain the formula
......@@ -851,7 +850,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
\[
\src(\alpha_i)=\trgt(\alpha_{i+1}).
\]
The source and target of alpha are given by
The source and target of $\alpha$ are given by
\[
\src(\alpha):=\src_0(\alpha_1)\text{ and
}\trgt(\alpha):=\trgt_0(\alpha_1).
......
......@@ -61,12 +61,18 @@ Consider the commutative square
\sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0).
\end{tikzcd}
\]
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
%% \[
%% \alpha_{\src_{n-1}(e_n)}=e_n=\alpha_{\src_{n-1}(e_n)}.
%% \]
\end{proof}
In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \ref{paragr:inclusionsphereglobe} that for every $n \geq 0$, 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\geq 0$, 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
\[
p : f(x)\to p \text{ if }n=0
p : f(x)\to a \text{ if }n=0
\]
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
\end{align*}
By universal property, this induces a map
\[
\Sigma^{X/a}_n \to \coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),a\right),
\coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),a\right) \to \Sigma^{X/a}_n,
\]
which is natural in $a$. A simple verification shows that it is a bijection.
\end{proof}
......@@ -491,7 +497,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
\[
\{i_n: \sS_{n-1} \to \sD_n \vert n \in \mathbb{N} \}
\]
is a set a generating folk cofibrations.
is a set of generating folk cofibrations.
From Lemmas \ref{lemma:sliceisfree} and \ref{lemma:basisofslice} we deduce
that for every object $a$ of $A$ and every $n \in \mathbb{N}$, the canonical square
\[
......
This diff is collapsed.
......@@ -25,7 +25,8 @@
The two main points to retain are:
\begin{description}
\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 $\langle 0 \,1 \cdots n\rangle$. 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
\[
{N_n : (n\Cat,\W_n^{\Th}) \to (\Psh{\Delta},\W_{\Delta})}
\]
and hence a morphism of op\nbd{}prederivators
\[
\overline{N_n} : \Ho(n\Cat^{\Th}) \to \Ho(\Psh{\Delta}).
\]
......@@ -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
\[
N_{n}c_{n}Q \overset{\alpha}{\longleftarrow} Q \overset{\gamma}{\longrightarrow} \mathrm{id}_{\Psh{\Delta}}
\]
......@@ -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
\[
\sD_1 \otimes C \overset{p\otimes i}{\longrightarrow} \sD_0 \otimes D \simeq D,
\sD_1 \otimes C \overset{p\otimes u}{\longrightarrow} \sD_0 \otimes D \simeq D,
\]
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 $A/b_0 \to A$.
\end{enumerate}
\end{proof}
......
......@@ -44,7 +44,7 @@ is poorly behaved. For example, \fi
\gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D)
\]
is fully faithful and its essential image consists of functors $F~:~\C~\to~\D$
that sends the morphisms of $\W$ to isomorphisms of $\D$.
that send the morphisms of $\W$ to isomorphisms of $\D$.
We shall always consider that $\C$ and $\ho(\C)$ have the same class of
objects and implicitly use the equality
......@@ -280,7 +280,7 @@ we shall use in the sequel.
which we call the op\nbd{}prederivator \emph{represented by $\C$}. For $u : A
\to B$ in $\CCat$,
\[
u^* : \C(A) \to \C(B)
u^* : \C(B) \to \C(A)
\]
is simply the functor induced from $u$ by pre-composition.
\end{example}
......@@ -317,7 +317,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\]
and every natural transformation $\begin{tikzcd}A \ar[r,bend
left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B
\ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ induces natural
\ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ induces a natural
transformation, again denoted by $\alpha^*$,
\[
\begin{tikzcd}
......@@ -360,7 +360,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\]
is nothing but the diagonal functor that sends an object $X$ of $\C$ to the
constant diagram with value $X$. Hence, the functor $p_{A!}$ is nothing but
the usual functor colimit functor of $A$-shaped diagrams
the usual colimit functor of $A$-shaped diagrams
\[
p_{A!} = \colim_A : \C(A) \to \C(e) \simeq \C.
\]
......@@ -433,7 +433,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{tikzcd}
\]
In particular, let $u : A \to B$ be a morphism $\CCat$ and $b$ an object of
In particular, let $u : A \to B$ be a morphism of $\CCat$ and $b$ an object of
$B$ seen as a morphism $b :e \to B$. We have a square
\[
\begin{tikzcd}
......@@ -444,7 +444,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
where :
\begin{itemize}[label=-]
\item $A/b$ is the category whose objects are pairs $(a, f : u(a) \to b)$ with
$a$ an object of $A$ and $p$ an arrow of $B$, and morphisms $(a,f) \to
$a$ an object of $A$ and $f$ an arrow of $B$, and morphisms $(a,f) \to
(a',f')$ are arrows $g : a \to a'$ of $A$ such that $f'\circ u(g) = f$,
\item $k : A/b \to A$ is the functor $(a,p) \mapsto a$,
\item $\phi$ is the natural transformation defined by $\phi_{(a,f)}:= f : u(a)
......@@ -474,7 +474,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\[
\sD(\amalg_{i \in I}A_i) \to \prod_{i \in I}\sD(A_i)
\]
is an equivalence categories. In particular, $\sD(\emptyset)$ is equivalent
is an equivalence of categories. In particular, $\sD(\emptyset)$ is equivalent
to the terminal category.
\item[Der 2)]\label{der2} For every small category $A$, the functor
\[
......@@ -578,7 +578,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{paragr}
\begin{example}
Let $F : \C \to \D$ be a functor. It induces a strict morphism at the level
Let $F : \C \to \C'$ be a functor. It induces a strict morphism at the level
of op\nbd{}prederivators, again denoted by $F$, where for every small
category $A$, the functor $F_A : \C(A) \to \C'(A)$ is induced by
post-composition. Similarly, every natural transformation induces a
......@@ -610,7 +610,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{example}
\begin{paragr}\label{paragr:canmorphismcolimit}
Let $\sD$ and $\sD'$ be op\nbd{}prederivators that admit left Kan extensions
and let $F : \sD \to \sD'$ morphism of op\nbd{}prederivators. For every $u :
and let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators. For every $u :
A \to B$, there is a canonical natural transformation
\[
u_!\, F_A \Rightarrow F_B\, u_!
......@@ -633,9 +633,9 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\]
\end{paragr}
\begin{definition}\label{def:cocontinuous}
Let $F : \sD \to \sD'$ morphism of op\nbd{}prederivators and suppose that
Let $F : \sD \to \sD'$ be a morphism of op\nbd{}prederivators and suppose that
$\sD$ and $\sD'$ both admit left Kan extensions. We say that $F$ is
\emph{cocontinuous} or \emph{left exact} if for every $u: A \to B$, the
\emph{cocontinuous}\footnote{Some authors also say \emph{left exact}.} if for every $u: A \to B$, the
canonical morphism
\[
u_! \, F_A \Rightarrow F_B \, u_!
......@@ -780,7 +780,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{paragr}
\begin{definition}\label{def:cocartesiansquare}
Let $\sD$ be an op\nbd{}prederivator. An object $X$ of $\sD(\square)$ is
\emph{cocartesian} if for every $Y$ object of $\sD(\square)$, the canonical
\emph{cocartesian} if for every object $Y$ of $\sD(\square)$, the canonical
map
\[
\Hom_{\sD(\square)}(X,Y) \to
......@@ -877,7 +877,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
Hence, for a homotopy cocomplete localizer $(\C,\W)$, a commutative square of
$\C$ is homotopy
cocartesian if and only if the bottom right apex of the square is the homotopy
colimit of upper left corner of the square. This hopefully justify the
colimit of the upper left corner of the square. This hopefully justifies the
terminology of ``cocartesian square''.
The previous proposition admits the following immediate corollary.
......@@ -974,7 +974,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{remark}
Even if Theorem \ref{thm:cisinskiI} tells us that (the homotopy
op\nbd{}prederivator of) a model category $(\M,\W,\Cof,\Fib)$ have homotopy
op\nbd{}prederivator of) a model category $(\M,\W,\Cof,\Fib)$ has homotopy
left Kan extensions, it is not generally true that for a small category $A$
the category of diagrams $\M(A)$ admits a model structure with the pointwise
weak equivalences as its weak equivalences. Hence, in general we cannot use
......@@ -1012,7 +1012,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\item the fibrations are the pointwise fibrations,
\item the cofibrations are those morphisms which have the left lifting property
to trivial fibrations. \end{itemize} Moreover, this model structure is
cofibrantly generated and a set of generating cofibration (resp.\ trivial
cofibrantly generated and a set of generating cofibrations (resp.\ trivial
cofibrations) is given by
\[
\{ f \otimes a : X \otimes a \to Y \otimes a \quad \vert \quad a \in
......@@ -1056,7 +1056,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
p_A^*
\end{tikzcd}
\]
is a Quillen adjunction. Since $\hocolim_A$ is the left derived of
is a Quillen adjunction. Since $\hocolim_A$ is the left derived functor of
$\colim_A$, we obtain the following immediate corollary of the previous
proposition.
\end{paragr}
......
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