@@ -135,7 +135,7 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re

\sD_n \ar[r,"j_n^-"']&\sS_{n}.

\end{tikzcd}

\]

is cartesian all of the four morphisms are monomorphisms. Since the

is cartesian and all four morphisms are monomorphisms. Since the

functor $\Hom_{\oo\Cat}(\Or_k,-)$ preserves limits, the square

\eqref{squarenervesphere} is a cartesian square of $\Set$ all of

whose four morphisms are monomorphisms. Hence, in order to prove

...

...

@@ -160,7 +160,21 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re

\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 $k=n$:] 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 $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If $x$ is a unit on a cell whose dimension is strictly lower than $n$, then everything works like in the case $k<n$. If not, this means that $x$ is a non-trivial $n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let $\gamma$ be a generator of $\Or_k$ of dimension $k-1$. We have $\varphi(\gamma)=\1^{k-1}_y$ with $y$ which is either a unit on a cell of dimension strictly lower than $n$, or a non-degenerate $n$\nbd{}cell of $\sS_n$ (if $k-1=n$, recall the convention that $\1^{k-1}_y=y$). In the first situation, $y$ is in the image of $j^+_n$ as in the case $k<n$, and thus, so is $\1^{k-1}_y$. In the second situation, this means \emph{a priori} that either $y=h_n^+$ or $y=h_n^-$. But we know that $\gamma$ is part of a composition that is equal to either the source or the target of $\alpha_k$ (see \ref{paragr:orientals}) and thus, $f(\gamma)$ is part of a composition that is equal to either the source or the target of $x=h^+_n$. Since no composition involving $h^-_n$ can be equal to $h^+_n$ (one could invoke the function introduced in \ref{prop:countingfunction}), this implies that $y=h_n^+$ and hence, $f(\gamma)$ is in the image of $j^+_n$. This goes for all generating cells of dimension $k-1$ of $\Or_k$ and we can recursively apply the same reasoning for generating cells of dimension $k-2$, then $k-3$ and so forth. Altogether, this proves that $\varphi$ factorizes through $j^+_n$. The case where $x=h^-_n$ and $\varphi$ factorizes through $j^-_n$ 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 $x$

a cell of $\sS_n$ of dimension non-greater than $n$. If $x$ is a unit on a

cell whose dimension is strictly lower than $n$, then everything works like

in the case $k<n$. If not, this means that $x$ is a non-trivial

$n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let

$\gamma$ be a generator of $\Or_k$ of dimension $k-1$. We have

$\varphi(\gamma)=\1^{k-1}_y$ with $y$ which is either a unit on a cell of

dimension strictly lower than $n$, or a non-degenerate $n$\nbd{}cell of

$\sS_n$ (if $k-1=n$, we use the convention that $\1^{k-1}_y=y$). In the

first situation, $y$ is in the image of $j^+_n$ as in the case $k<n$, and

thus, so is $\1^{k-1}_y$. In the second situation, this means \emph{a

priori} that either $y=h_n^+$ or $y=h_n^-$. But we know that $\gamma$ is

part of a composition that is equal to either the source or the target of

$\alpha_k$ (see \ref{paragr:orientals}) and thus, $y$ is part of a composition that is equal to either the source or the target of $x=h^+_n$. Since no composition involving $h^-_n$ can be equal to $h^+_n$ (one could invoke the function introduced in \ref{prop:countingfunction}), this implies that $y=h_n^+$ and hence, $f(\gamma)$ is in the image of $j^+_n$. This goes for all generating cells of dimension $k-1$ of $\Or_k$ and we can recursively apply the same reasoning for generating cells of dimension $k-2$, then $k-3$ and so forth. Altogether, this proves that $\varphi$ factorizes through $j^+_n$. The case where $x=h^-_n$ and $\varphi$ factorizes through $j^-_n$ is symmetric.

\end{description}

\end{proof}

From these two lemmas, follows the important proposition below.

...

...

@@ -275,7 +289,7 @@ higher than $1$.

(a',p)&\mapsto a'.

\end{aligned}

\]

This is 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{}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

\[

f/a : X/a \to A/a

\]

...

...

@@ -286,7 +300,19 @@ higher than $1$.

A/a \ar[r,"\pi_{a}"]&A.

\end{tikzcd}

\]

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(\trgt_0(x))\to a$. When $n>1$, the source and target of such an $n$\nbd{}cell are given by

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

\]

and

\[

p : f(\trgt_0(x))\to a \text{ if }n>0.

\]

\emph{From now on, let us use the convention that $\trgt_0(x)=x$ when $x$ is a

$0$\nbd{}cell of $X$}.

When $n>0$, the source and target of an $n$\nbd{}cell $(x,p)$ of $X/a$ are given by

\[

\src((x,p))=(\src(x),p)\text{ and }\trgt((x,p))=(\trgt(x),p).

\]

...

...

@@ -321,7 +347,7 @@ higher than $1$.

\[

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

\]

This map is natural in $X$ in the following sense. Let $f' : X' \to A$ be another $\oo$\nbd{}functor and let

Let $f' : X' \to A$ be another $\oo$\nbd{}functor and let

\[

\begin{tikzcd}

X \ar[rr,"g"]\ar[dr,"f"']&& X' \ar[dl,"f'"]\\

...

...

@@ -383,7 +409,7 @@ higher than $1$.

\]

is commutative. This proves the existence part of the universal property.

Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangles commute for every object $a$ of $A$ and let $x$ be a $n$\nbd{}cell of $X$. Since the triangle

Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangles commute for every object $a$ of $A$ and let $x$ be an$n$\nbd{}cell of $X$. Since the triangle

\[

\begin{tikzcd}

X/f(\trgt_0(x))\ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r]& X \ar[d,"\phi'"]\\

...

...

@@ -503,8 +529,8 @@ Beware that in the previous corollary, we did \emph{not} suppose that $X$ was fr

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

@@ -572,7 +598,7 @@ We now recall an important theorem due to Thomason.

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

Let $A$ be a $1$\nbd{}category. The canonical map

\[

\hocolim_{a \in A}(A/a)\to A

\hocolim^{\Th}_{a \in A}(A/a)\to A

\]

induced by the co-cone $(A/a \to A)_{a \in\Ob(A)}$, is an isomorphism of $\ho(\Cat^{\Th})$.

\end{corollary}

...

...

@@ -591,14 +617,14 @@ We now recall an important theorem due to Thomason.

\[

\int_{a \in A}A/a \to\int_{a \in A}k_{\sD_0}

\]

is a Thomason equivalence. An immediate computation shows that \[\int_{a \in A}k_{\sD_0}\simeq A.\] From the second part of Theorem \ref{thm:Thomason}, we have that

is a Thomason equivalence and an immediate computation shows that \[\int_{a \in A}k_{\sD_0}\simeq A.\] From the second part of Theorem \ref{thm:Thomason}, we have that

\[

\hocolim_{a \in A}(A/a)\simeq A.

\hocolim^{\Th}_{a \in A}(A/a)\simeq A.

\]

A thorough analysis of all the isomorphisms involved shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in\Ob(A)}$.

\end{proof}

\begin{remark}

It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}categories), we have \[\hocolim^{\Th}_{a \in A}(X/a)\simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.

It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}categories), we have \[\hocolim^{\Th}_{a \in A}(X/a)\simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation.

\end{remark}

Putting all the pieces together, we are now able to prove the awaited Theorem.

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

Now let $\Delta_{\leq2}$ be the full subcategory of $\Delta$ spanned by

$[0]$, $[1]$ and $[2]$ and let $i : \Delta_{\leq2}\to\Delta$ be the

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

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

\Psh{\Delta_{\leq2}}\to\Psh{\Delta}$. Recall that the nerve of a (small) category is $2$-coskeletal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D)\to i_* i^*(N_1(D))$ is an isomorphism of simplicial sets. In particular, we have

@@ -209,7 +209,12 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

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

For every $1\leq n \leq\oo$, the localizer $(n\Cat^{\Th},\W_n^{\Th})$ is homotopy cocomplete (Definition \ref{def:cocompletelocalizer}).

\end{corollary}

Another consequence of Gagna's theorem is the following corollary.

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

cocartesian square'' 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

corollary.

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

For every $1\leq n \leq\oo$, the class $\W_n^{\Th}$ is saturated (\ref{paragr:loc}).

\end{corollary}

...

...

@@ -557,8 +562,11 @@ For later reference, we put here the following trivial but important lemma, whos

\[

\gamma^{\folk} : \oo\Cat\to\Ho(\oo\Cat^{\folk})

\]

for the localization morphism.

It follows from the previous theorem and Theorem \ref{thm:cisinskiI} that the localizer $(\oo\Cat,\W_{\oo}^{\folk})$ is homotopy cocomplete.

for the localization morphism. It follows from the previous theorem and

Theorem \ref{thm:cisinskiI} that the localizer $(\oo\Cat,\W_{\oo}^{\folk})$

is homotopy cocomplete. We will speak of ``folk homotopy

colimits'' and ``folk homotopy cocartesian squares'' for homotopy colimits

and homotopy cocartesian squares in this localizer.

\end{paragr}

\begin{paragr}\label{paragr:folktrivialfib}

Using the set $\{i_n : \sS_{n-1}\to\sD_n \vert n \in\mathbb{N}\}$ of generating folk cofibrations, we obtain that an $\oo$\nbd{}functor $F : C \to D$ is a \emph{folk trivial fibration} when: