Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to\sD_0$ the canonical morphism to the terminal object of $\sD_0$.

Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to\sD_0$for the canonical morphism to the terminal object of $\sD_0$.

\begin{definition}\label{def:contractible}

An $\oo$\nbd{}category $C$ is \emph{oplax contractible} when the canonical morphism $p_C : C \to\sD_0$ is an oplax homotopy equivalence (Definition \ref{def:oplaxhmtpyequiv}).

It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left 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.

\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.

...

...

@@ -137,11 +137,13 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re

\]

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

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

\eqref{squarenervesphere} is a cartesian square of $\Set$ and all of whose four morphisms are

monomorphisms.

Hence, what we need to show is that for every $k \geq0$ and every

$\oo$\nbd{}functor $\varphi : \Or_k \to\sS_{n}$, there exists an

$\oo$\nbd{}functor $\varphi' : \Or_k \to\sD_n$ such that either $j_n^+\circ\varphi ' =\varphi$ or $j_n^-\circ\varphi' =\varphi$.

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

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

that square \eqref{squarenervesphere} is cocartesian, we only need

to show that for every $k \geq0$ and every $\oo$\nbd{}functor

$\varphi : \Or_k \to\sS_{n}$, there exists an $\oo$\nbd{}functor

$\varphi' : \Or_k \to\sD_n$ such that either $j_n^+\circ\varphi '

=\varphi$ or $j_n^-\circ\varphi' =\varphi$.

%% Notice now that the morphisms $j_n^+$ and $j_n^-$ trivially satisfy the following

%% properties:

...

...

@@ -158,7 +160,7 @@ 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 the dimension of $x$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 $k-1$. Either we have $\varphi(\gamma)=1^n_y$ for $y$ a cell of dimension strictly lower than $n$, or we have that $\varphi(\gamma)$ is a non-degenerate $n$\nbd{}cell of $\sS_n$. In the first situation, $y$ is in the image of $j^+_n$ as in the case $k<n$, and thus, so is $1^n_y$. 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 $k-1=n$, 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.

\end{description}

\end{proof}

From these two lemmas, follows the important proposition below.

...

...

@@ -186,7 +188,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an

\end{tikzcd}

\]

is Thomason homotopy cocartesian for every $n\geq0$. Finally, since $i_n : \sS_{n-1}\to\sD_{n}$ is

a folk cofibration and $\sS_{n-1}$ and $\sD_{n}$ are folk cofibrant for every $n\geq0$, we deduce the result from Corollary \ref{cor:usefulcriterion} and an immediate induction. The base case being simply that $\sS_{-1}=\emptyset$ is obviously \good{}.

a folk cofibration and $\sS_{n-1}$ and $\sD_{n}$ are folk cofibrant for every $n\geq0$, we deduce the desired result from Corollary \ref{cor:usefulcriterion} and an immediate induction. The base case being simply that $\sS_{-1}=\emptyset$ is obviously \good{}.

\end{proof}

\begin{paragr}

The previous proposition implies what we claimed in Paragraph \ref{paragr:prelimcriteriongoodcat}, which is that the morphism of op-prederivators

...

...

@@ -204,16 +206,17 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an

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

\end{tikzcd}

\]

where the map $\sD_2\to B^2\mathbb{N}$ points the unique generating $2$-cell of $B^2\mathbb{N}$ and

$\sD_0\to B^2\mathbb{N}$ points to the only object of $B^2\mathbb{N}$. It is easily checked that this

square is cocartesian and since $\sS_1$, $\sD_0$ and $\sD_2$ are free and

$i_2$ is a cofibration for the canonical model structure, the square is also

homotopy cocartesian with respect to folk weak equivalences. If $\J$ was

homotopy cocontinuous, then this square would also be homotopy cocartesian

with respect to Thomason equivalences. Since we know that $\sS_1$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}.

where the map $\sD_2\to B^2\mathbb{N}$ points the unique generating

$2$-cell of $B^2\mathbb{N}$ and $\sD_0\to B^2\mathbb{N}$ points to

the only object of $B^2\mathbb{N}$. Since $\sS_1$, $\sD_0$ and $\sD_2$ are

free and $i_2$ is a folk cofibration,

the square is also folk homotopy cocartesian. If $\J$ was homotopy

cocontinuous, then this square would also be Thomason homotopy

cocartesian. Since we know that $\sS_1$, $\sD_0$ and $\sD_2$ are

\good{}, this would imply that $B^2\mathbb{N}$ is \good{}.

From Proposition \ref{prop:spheresaregood}, we also deduce the proposition

below which gives a criterion to detect \good{}$\oo$\nbd{}category when we

below which gives a criterion to detect \good{}$\oo$\nbd{}categories when we

already know that they are free.

\end{paragr}

\begin{proposition}

...

...

@@ -231,20 +234,22 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an

is homotopy cocartesian with respect to Thomason equivalences, then $C$ is \good{}.

\end{proposition}

\begin{proof}

Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd{}categories

$\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant and \good{}, it follows

from Corollary \ref{cor:usefulcriterion} that all $\sk_k(C)$ are \good{}. The

result follows then from Lemma \ref{lemma:filtration} and Proposition \ref{prop:sequentialhmtpycolimit}.

Since the morphisms $i_k$ are folk cofibration 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

follows then from Lemma \ref{lemma:filtration} and Proposition

\ref{prop:sequentialhmtpycolimit}.

\end{proof}

\section{The miraculous case of 1-categories}

Recall that the terms \emph{1-category} and \emph{(small) category} are

synonymous. While we have used the latter one more often so far, in this section

we will mostly use the former one. As usual, the canonical functor $\iota_1 :

\Cat\to\oo\Cat$ is treated as an inclusion functor and hence we always consider

$(1-)$categories as particular cases of $\oo$\nbd{}categories.

$1$\nbd{}categories as particular cases of $\oo$\nbd{}categories.

The goal of what follows is to show that every $1$-category is \good{}. In order to do that, we will prove that every 1-category

is a canonical colimit of contractible $1$-categories and that this colimit is

The goal of what follows is to show that every $1$\nbd{}category is \good{}. In order to do that, we will prove that every 1-category

is a canonical colimit of contractible $1$\nbd{}categories and that this colimit is

homotopic both

with respect to the folk weak equivalences and with respect to the Thomason equivalences.

We call the reader's attention to an important subtlety here: even though the

...

...

@@ -258,67 +263,63 @@ then $P$ is not necessarily a $1$\nbd{}category. In particular, polygraphic

homology groups of a $1$\nbd{}category need \emph{not} be trivial in dimension

higher than $1$.

\begin{paragr}

Recall that for an object $a_0$ of an $\oo$\nbd{}category $A$, we denote by $A/a_0$ the slice $\oo$\nbd{}category over $a_0$ (Paragraph \ref{paragr:slices}). When $A$ is a $1$-category, $A/a_0$ is also a $1$-category whose description is as follows:

\begin{itemize}[label=-

]

\item an object of $A/a_0$ is a pair $(a, p : a \to a_0)$ where $a$ is an object of $A$ and $p$ is an arrow of $A$,

\item an arrow $(a,p)\to(a',p')$ of $A/a_0$ is an arrow $ q : a \to a'$ of $A$ such that $p'\circ q = p$.

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 $p'\circ q = p$,

\end{itemize}

We denote by

\[

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

\[

\begin{aligned}

\pi_{a_0} : A/a_0&\to A \\

(a,p)&\mapsto a

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

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

\end{aligned}

\]

the canonical forgetful functor.

Recall also that given an $\oo$\nbd{}functor $f : X \to A$ and an object $a_0$

of $A$, we have defined the $\oo$\nbd{}category $A/a_0$ and the $\oo$\nbd{}functor

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

\[

f/a_0 : X/a_0\to A//a_0

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

\]

as the following pullback

\[

\begin{tikzcd}

X/a_0\ar[r]\ar[dr, phantom, "\lrcorner", very near start]\ar[d,"f/a_0"']& X \ar[d,"f"]\\

A/a_0\ar[r,"\pi_{a_0}"]&A.

X/a \ar[r]\ar[dr, phantom, "\lrcorner", very near start]\ar[d,"f/a"']& X \ar[d,"f"]\\

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

\end{tikzcd}

\]

When $A$ is a $1$-category, the $n$-cells of $X/a_0$ 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(\trgt_0(x))\to a$. When $n>1$, the source and target of such an $n$\nbd{}cell are given by

\[

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

\]

Moreover, the $\oo$\nbd{}functor $f/a_0$ is described as

Moreover, the $\oo$\nbd{}functor $f/a$ is described as

\[

(x,p)\mapsto(f(x),p),

\]

and the canonical $\oo$\nbd{}functor $X \to X/a_0$ as

and the canonical $\oo$\nbd{}functor $X \to X/a$ as

\[

(x,p)\mapsto x.

\]

\end{paragr}

\begin{paragr}\label{paragr:unfolding}

Let $f : X \to A$ be an $\oo$\nbd{}functor with $A$ a $1$-category. Every arrow $\beta : a_0\to a_0'$ induces an $\oo$\nbd{}functor

Let $f : X \to A$ be an $\oo$\nbd{}functor with $A$ a $1$\nbd{}category. Every arrow $\beta : a \to a'$ of $A$ induces an $\oo$\nbd{}functor