Commit 8e07074f authored by Leonard Guetta's avatar Leonard Guetta
Browse files

dlkfjqsdlkqsjdh

parent d97a025b
......@@ -744,7 +744,7 @@ For any $n \geq 0$, consider the following cocartesian square
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd},
\]
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trival $1$\nbd-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong deformation retract, hence a co-universal Thomason equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)} \to A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is Thomason homotopy cocartesian. Now, the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from Corollary \ref{cor:usefulcriterion} that $A_{(1,n)}$ is \good{}. Finally, since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a point, the fact that the previous square is Thomason homotopy cocartesian implies that $A_{(1,n)}$ has the homotopy type of a point.
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trival $1$\nbd{}cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong deformation retract, hence a co-universal Thomason equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)} \to A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is Thomason homotopy cocartesian. Now, the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from Corollary \ref{cor:usefulcriterion} that $A_{(1,n)}$ is \good{}. Finally, since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a point, the fact that the previous square is Thomason homotopy cocartesian implies that $A_{(1,n)}$ has the homotopy type of a point.
Similarly, for any $m \geq 0$, by considering the cocartesian square
\[
......@@ -754,7 +754,7 @@ For any $n \geq 0$, consider the following cocartesian square
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd},
\]
where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non trivial $1$\nbd-cell of $\Delta_1$ the source of the generating $2$\nbd-cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has the homotopy type of a point.
where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non trivial $1$\nbd{}cell of $\Delta_1$ the source of the generating $2$\nbd{}cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has the homotopy type of a point.
Now, let $m\geq 0$ and $n > 0$ and consider the cocartesian square
\[
......
\chapter{Homology of contractible $\omega$-categories its consequences}
\section{Contractible $\oo$-categories}
Recall that for any $\oo$\nbd-category $C$, we write $p_C : C \to \sD_0$ the canonical morphism to the terminal object of $\sD_0$.
Recall that for any $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ 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}).
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}).
%% there exists an object $x_0$ of $X$ and an oplax transformation
%% \[
......@@ -16,11 +16,11 @@ Recall that for any $\oo$\nbd-category $C$, we write $p_C : C \to \sD_0$ the can
%% \begin{paragr}
%% In other words, an $\oo$\nbd-category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason weak equivalence. In particular, we have the following lemma.
%% In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason weak equivalence. In particular, we have the following lemma.
%% \end{paragr}
%% \begin{lemma}\label{lemma:hmlgycontractible}
%% Let $X$ be a contractible $\oo$\nbd-category. The morphism of $\ho(\Ch)$
%% Let $X$ be a contractible $\oo$\nbd{}category. The morphism of $\ho(\Ch)$
%% \[
%% \sH(X) \to \sH(\sD_0)
%% \]
......@@ -47,7 +47,7 @@ Recall that for any $\oo$\nbd-category $C$, we write $p_C : C \to \sD_0$ the can
%% We can now prove the main result of this section.
\begin{proposition}\label{prop:contractibleisgood}
Every oplax contractible $\oo$\nbd-category $C$ is \good{} and we have
Every oplax contractible $\oo$\nbd{}category $C$ is \good{} and we have
\[
\sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}
\]
......@@ -64,11 +64,11 @@ Consider the commutative square
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, a simple explicit computation 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.
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 slices $\oo$\nbd{}category (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.
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}
\begin{proof}
This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}.
......@@ -78,7 +78,7 @@ We end this section with an important result on slices $\oo$\nbd-category (Parag
\todo{Rappeler définitions par récurrence des globes et sphères.}
\end{paragr}
\begin{lemma}\label{lemma:globescontractible}
For every $n \in \mathbb{N}$, the $\oo$\nbd-category $\sD_n$ is contractible.
For every $n \in \mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is contractible.
\end{lemma}
\begin{proof}
\todo{À écrire}
......@@ -99,11 +99,11 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}.
\end{proof}
From these two lemmas, follows the important proposition below.
\begin{proposition}\label{prop:spheresaregood}
For every $n \geq -1$, the $\oo$\nbd-category $\sS_n$ is \good{}.
For every $n \geq -1$, the $\oo$\nbd{}category $\sS_n$ is \good{}.
\end{proposition}
\begin{proof}
We proceed by induction on $n$. When $n=-1$, it is trivial to check that the
empty $\oo$\nbd-category is \good{}. Now, since $i_n : \sS_n \to \sD_{n+1}$ is
empty $\oo$\nbd{}category is \good{}. Now, since $i_n : \sS_n \to \sD_{n+1}$ is
a folk cofibration and $\sS_{n}$ and $\sD_{n}$ are folk cofibrant, it follows
from Lemma \ref{lemma:hmtpycocartesianreedy} that the cocartesian square
\begin{equation}\label{square}
......@@ -132,7 +132,7 @@ From these two lemmas, follows the important proposition below.
cocartesian square
\[
\begin{tikzcd}
\sS_2 \ar[r] \ar[d,"i_2"] & \sD_0 \ar[d] \\
\sS_1 \ar[r] \ar[d,"i_2"] & \sD_0 \ar[d] \\
\sD_2 \ar[r] & B^2\mathbb{N},
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
......@@ -143,16 +143,16 @@ From these two lemmas, follows the important proposition below.
$i_2$ 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 $\sD_2$, $\sD_0$ and $\sD_2$ are \good{}, this would imply that $B^2\mathbb{N}$ is \good{}.
with respect to Thomason equivalences. Since we know that $\sS_A$, $\sD_0$ 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{}category when we
already know that they are free. Note that it seems hard
to use in practise and we will only use it for theoretical purposes.
\end{paragr}
\begin{proposition}
Let $C$ be a free $\oo$\nbd-category and for every $k \in \mathbb{N}$ let
$\Sigma_k$ be its $k$\nbd-basis. If for every $k \in \mathbb{N}$, the following
Let $C$ be a free $\oo$\nbd{}category and for every $k \in \mathbb{N}$ let
$\Sigma_k$ be its $k$\nbd{}basis. If for every $k \in \mathbb{N}$, the following
cocartesian square (see \ref{def:nbasis})
\[
\begin{tikzcd}[column sep=large]
......@@ -165,7 +165,7 @@ From these two lemmas, follows the important proposition below.
is homotopy cocartesian with respect to Thomason equivalences, then $X$ is \good{}.
\end{proposition}
\begin{proof}
Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd-categories
Since the morphisms $i_k$ are folk cofibration and the $\oo$\nbd{}categories
$\sS^{k-1}$ and $\sS^{k}$ 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}.
......@@ -175,24 +175,24 @@ 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.
$(1-)$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
homotopic both
with respect to folk weak equivalences and and with respect to Thomason equivalences.
We call the reader's attention to an important subtlety here: even though the
desired result only refers to $1$\nbd-categories, we have to work in the setting
of $\oo$\nbd-categories. This can be explained from the fact that if we take a
cofibrant resolution of a $1$\nbd-category $C$ in the folk model structure on $\oo\Cat$
desired result only refers to $1$\nbd{}categories, we have to work in the setting
of $\oo$\nbd{}categories. This can be explained from the fact that if we take a
cofibrant resolution of a $1$\nbd{}category $C$ in the folk model structure on $\oo\Cat$
\[
P \to C,
\]
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
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:
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 $A$,
......@@ -207,8 +207,8 @@ higher than $1$.
\]
the canonical forgetful functor.
Recall also that given an $\oo$\nbd-functor $f : X \to A$ and an object $a_0$
of $A$, we defined the $\oo$\nbd-category $A/a_0$ and the $\oo$\nbd-functor
Recall also that given an $\oo$\nbd{}functor $f : X \to A$ and an object $a_0$
of $A$, we defined the $\oo$\nbd{}category $A/a_0$ and the $\oo$\nbd{}functor
\[
f/a_0 : X/a_0 \to A//a_0
\]
......@@ -219,21 +219,21 @@ higher than $1$.
A/a_0 \ar[r,"\pi_{a_0}"] &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
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
\[
\src((x,p))=(\src(x),p) \text{ and } \trgt((x,p))=(\trgt(x),p).
\]
Moreover, the $\oo$\nbd-functor $f/a_0$ is described as
Moreover, the $\oo$\nbd{}functor $f/a_0$ 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 $X \to X/a_0$ 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. Any 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 $1$-category. Any arrow $\beta : a_0 \to a_0'$ induces an $\oo$\nbd{}functor
\begin{align*}
X/\beta : X/a_0 &\to X/{a_0'} \\
(x,p) & \mapsto (x,\beta \circ p),
......@@ -254,19 +254,19 @@ higher than $1$.
\[
\colim_{a_0 \in A} X/{a_0} \to X.
\]
This map is natural in $X$ in the following sense. Let $f' : X' \to A$ be another $\oo$\nbd-functor and let
This map is natural in $X$ in the following sense. Let $f' : X' \to A$ be another $\oo$\nbd{}functor and let
\[
\begin{tikzcd}
X \ar[rr,"g"] \ar[dr,"f"'] && X' \ar[dl,"f'"] \\
&A&
\end{tikzcd}
\]
be a commutative triangle in $\oo\Cat$. For every object $a_0$ of $A$, there is an $\oo$\nbd-functor $g/a_0$ defined as
be a commutative triangle in $\oo\Cat$. For every object $a_0$ of $A$, there is an $\oo$\nbd{}functor $g/a_0$ defined as
\begin{align*}
X/a_0 &\to X'/a_0 \\
(x,p) &\mapsto (g(x),p),
\end{align*}
which induces an $\oo$\nbd-functor
which induces an $\oo$\nbd{}functor
\[
\colim_{a_0 \in A}(X/a_0) \to \colim_{a_0 \in A}(X'/a_0)
\]
......@@ -280,7 +280,7 @@ higher than $1$.
is commutative.
\end{paragr}
\begin{lemma}\label{lemma:colimslice}
Let $f : X \to A$ be an $\oo$\nbd-functor such that $A$ is a $1$-category. The canonical arrow
Let $f : X \to A$ be an $\oo$\nbd{}functor such that $A$ is a $1$-category. The canonical arrow
\[
\colim_{a_0 \in A}(X/a_0) \to X
\]
......@@ -296,16 +296,16 @@ higher than $1$.
\[
(g_{a_0} : X/a_0 \to C)_{a_0 \in \Ob(A)}
\]
be another cocone and let $x$ be a $n$\nbd-arrow of $X$. Notice that the pair
be another cocone and let $x$ be a $n$\nbd{}arrow of $X$. Notice that the pair
\[
(x,1_{f(\trgt_0(x))})
\]
is a $n$\nbd-arrow of $X/f(\trgt_0(x))$. We leave it to the reader to prove that the formula
is a $n$\nbd{}arrow of $X/f(\trgt_0(x))$. We leave it to the reader to prove that the formula
\begin{align*}
\phi : X &\to C \\
x &\mapsto g_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))}).
\end{align*}
defines an $\oo$\nbd-functor. This proves the existence part of the universal property.
defines an $\oo$\nbd{}functor. This proves the existence part of the universal property.
It is straightforward to check that for every $a_0 \in \Ob(A)$ the triangle
\[
......@@ -314,7 +314,7 @@ higher than $1$.
& C
\end{tikzcd}
\]
is commutative. Now let $\phi' : X \to C$ be another $\oo$\nbd-functor that makes the previous triangles commute and let $x$ be a $n$\nbd-arrow of $X$. Since the triangle
is commutative. Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangles commute and let $x$ be a $n$\nbd{}arrow of $X$. Since the triangle
\[
\begin{tikzcd}
X/f(\trgt_0(x)) \ar[dr,"g_{f(\trgt_0(x))}"']\ar[r] & X \ar[d,"\phi'"] \\
......@@ -336,11 +336,11 @@ higher than $1$.
We now proceed to prove that this colimit is homotopic with respect to
folk weak equivalences.
\end{paragr}
Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd-functor $f : X \to A$ with $A$ a $1$\nbd-category.
Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}functor $f : X \to A$ with $A$ a $1$\nbd{}category.
\begin{lemma}\label{lemma:sliceisfree}
If $X$ is free, then for every object $a_0$ of $A$, the $\oo$\nbd-category
$X/a_0$ is free. More precisely, if $\Sigma^X_n$ is the $n$\nbd-basis of $X$,
then the $n$\nbd-basis of $X/a_0$ is the set
If $X$ is free, then for every object $a_0$ of $A$, the $\oo$\nbd{}category
$X/a_0$ is free. More precisely, if $\Sigma^X_n$ is the $n$\nbd{}basis of $X$,
then the $n$\nbd{}basis of $X/a_0$ is the set
\[
\Sigma^{X/a_0}_n := \{(x,p) \in (X/a_0)_n \vert x \in \Sigma^X_n \}.
\]
......@@ -349,7 +349,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd-funct
It is immediate to check that for every object $a_0$ of $A$, the canonical
forgetful functor $\pi_{a_0} : A/a_0 \to A$ is a Conduché functor (see Section
\ref{section:conduche}). Hence, from Lemma \ref{lemma:pullbackconduche} we
know that $X/a_0 \to X$ is a discrete $\oo$\nbd-functor. The result follows
know that $X/a_0 \to X$ is a discrete $\oo$\nbd{}functor. The result follows
then from Theorem \ref{thm:conduche}.
\end{proof}
\begin{paragr}
......@@ -383,7 +383,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd-funct
which is natural in $a_0$. A simple verification shows that it is a bijection.
\end{proof}
\begin{proposition}\label{prop:sliceiscofibrant}
Let $A$ a $1$\nbd-category, $X$ a free $\oo$\nbd-category and $f : X \to A$ be an $\oo$\nbd-functor. The functor
Let $A$ a $1$\nbd{}category, $X$ a free $\oo$\nbd{}category and $f : X \to A$ be an $\oo$\nbd{}functor. The functor
\begin{align*}
A &\to \oo\Cat \\
a_0 &\mapsto X/a_0
......@@ -411,8 +411,8 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd-funct
projective model structure on $\oo\Cat(A)$ and from Lemma
\ref{lemma:filtration} that $X/- : A \to \oo\Cat$ is cofibrant.
\end{proof}
\begin{corollary}\label{cor:folkhmptycol}
Let $A$ be a $1$\nbd-category and $f : X \to A$ an $\oo$\nbd-functor. The canonical arrow of $\ho(\oo\Cat^{\folk})$
\begin{corollary}\label{cor:folkhmtpycol}
Let $A$ be a $1$\nbd{}category and $f : X \to A$ an $\oo$\nbd{}functor. The canonical arrow of $\ho(\oo\Cat^{\folk})$
\[
\hocolim^{\folk}_{a_0 \in A}(X/a_0) \to X,
\]
......@@ -519,7 +519,7 @@ We now recall an important Theorem due to Thomason.
A thorough analysis of all the isomorphisms involved (\todo{détailler ou ref à Maltsiniotis}) 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 any functor $f : X \to A$ ($X$ and $A$ being $1$-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}, would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd-categorical analoguous 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 any functor $f : X \to A$ ($X$ and $A$ being $1$-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}, would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analoguous of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.
\end{remark}
Putting all the pieces together, we are now able to prove the awaited Theorem.
\begin{theorem}
......
This diff is collapsed.
......@@ -43,7 +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 composite 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 composite 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.
\end{description}
Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight of the $(n-1)$-cell corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ (see \ref{paragr:weight}) 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:
......@@ -350,7 +350,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\]
for each cell $x$ of $B$) defines an oplax transformation from $g \circ u$ to $g \circ v$ (resp. $u \circ f$ to $v\circ f$) that we denote $g\star \alpha$ (resp. $\alpha \star f$).
More abstractly, if $\alpha$ is seen as an $\oo$\nbd-functor $\sD_1 \otimes C \to D$, then $g \star \alpha$ (resp.\ $\alpha \star f)$ corresponds to the $\oo$\nbd-functor obtained as the following composition
More abstractly, if $\alpha$ is seen as an $\oo$\nbd{}functor $\sD_1 \otimes C \to D$, then $g \star \alpha$ (resp.\ $\alpha \star f)$ corresponds to the $\oo$\nbd{}functor obtained as the following composition
\[
\sD_1 \otimes C \overset{\alpha}{\longrightarrow} D \overset{f}{\longrightarrow} E
\]
......@@ -360,14 +360,14 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\]
\end{paragr}
\begin{remark}
All the above descriptions of oplax transformations can be easily dualized for \emph{lax transformations} (that is to say $1$\nbd-cells of the $\oo$\nbd-category $\underline{\hom}_{\mathrm{lax}}(X,Y)$ for some $\oo$-categories $X$ and $Y$). Habit is the only reason why we put emphasis on oplax transformations rather than lax transformations.
All the above descriptions of oplax transformations can be easily dualized for \emph{lax transformations} (that is to say $1$\nbd{}cells of the $\oo$\nbd{}category $\underline{\hom}_{\mathrm{lax}}(X,Y)$ for some $\oo$-categories $X$ and $Y$). Habit is the only reason why we put emphasis on oplax transformations rather than lax transformations.
\end{remark}
\section{Homotopy equivalences and deformation retracts}
\begin{paragr}\label{paragr:hmtpyequiv}
Let $C$ and $D$ be two $\oo$\nbd-categories and consider the smallest equivalence relation on the set $\Hom_{\oo\Cat}(C,D)$ such that two $\oo$\nbd-functors from $C$ to $D$ are equivalent if there is an oplax direction between them (in any direction). Let us say that two $\oo$-functors $u, v : C \to D$ are \emph{oplax homotopic} if there are equivalent under this equivalence relation.
Let $C$ and $D$ be two $\oo$\nbd{}categories and consider the smallest equivalence relation on the set $\Hom_{\oo\Cat}(C,D)$ such that two $\oo$\nbd{}functors from $C$ to $D$ are equivalent if there is an oplax direction between them (in any direction). Let us say that two $\oo$-functors $u, v : C \to D$ are \emph{oplax homotopic} if there are equivalent under this equivalence relation.
\end{paragr}
\begin{definition}\label{def:oplaxhmtpyequiv}
An $\oo$\nbd-functor $u : C \to D$ is an \emph{oplax homotopy equivalence} if there exists an $\oo$\nbd-functor $v : D \to C$ such that $u\circ v$ is oplax homotopic to $\mathrm{id}_D$ and $v\circ u$ is oplax homotopic to $\mathrm{id}_C$.
An $\oo$\nbd{}functor $u : C \to D$ is an \emph{oplax homotopy equivalence} if there exists an $\oo$\nbd{}functor $v : D \to C$ such that $u\circ v$ is oplax homotopic to $\mathrm{id}_D$ and $v\circ u$ is oplax homotopic to $\mathrm{id}_C$.
\end{definition}
In the following lemma, we denote by $\gamma : \oo\Cat \to \ho(\oo\Cat^{\Th})$ the localization functor with respect to the Thomason equivalences.
\begin{lemma}\label{lemma:oplaxloc}
......@@ -497,7 +497,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\begin{remark}
All the results we have seen in this section are still true if we replace ``oplax'' by ``lax'' everywhere.
\end{remark}
\section{Equivalence of $\omega$-categories and the folk model structure}
\section{Equivalences of $\omega$-categories and the folk model structure}
\begin{paragr}\label{paragr:ooequivalence}
Let $C$ be an $\omega$-category. We define the equivalence relation $\sim_{\omega}$ on the set $C_n$ by co-induction on $n \in \mathbb{N}$. For $x, y \in C_n$, we have $x \sim_{\omega} y $ when:
\begin{itemize}
......@@ -523,10 +523,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{example}
For later reference, we put here the following trivial but important lemma, whose proof is ommited.
\begin{lemma}
Let $F : C \to D$ be an $\oo$\nbd-functor and $x$,$y$ be $n$-cells of $C$ for some $n \geq 0$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$.
Let $F : C \to D$ be an $\oo$\nbd{}functor and $x$,$y$ be $n$-cells of $C$ for some $n \geq 0$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$.
\end{lemma}
\begin{definition}\label{def:eqomegacat}
An $\omega$-functor $D : C \to D$ is an \emph{equivalence of $\oo$\nbd-categories} when:
An $\omega$-functor $D : C \to D$ is an \emph{equivalence of $\oo$\nbd{}categories} when:
\begin{itemize}
\item[-] for every $y \in D_0$, there exists $x \in C_0$ such that
\[F(x)\sim_{\omega}y,\]
......@@ -538,7 +538,7 @@ For later reference, we put here the following trivial but important lemma, whos
\end{itemize}
\end{definition}
\begin{example}\label{example:equivalencecategories}
If $C$ and $D$ are (small) categories seen as $\oo$-categories, then a functor $F : C \to D$ is an equivalence of $\oo$\nbd-categories if and only if it is fully faithful and essentially surjective, i.e.\ an equivalence of categories.
If $C$ and $D$ are (small) categories seen as $\oo$-categories, then a functor $F : C \to D$ is an equivalence of $\oo$\nbd{}categories if and only if it is fully faithful and essentially surjective, i.e.\ an equivalence of categories.
\end{example}
For the next theorem, recall that the canonical maps $i_n : \sS_{n-1} \to \sD_n$ for $n \geq 0$ have been defined in \ref{paragr:defglobe}.
\begin{theorem}\label{thm:folkms}
......@@ -550,7 +550,7 @@ For later reference, we put here the following trivial but important lemma, whos
\begin{paragr}\label{paragr:folkms}
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.
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.
Similarly to the Thomason case (see \ref{paragr:notationthom}), we will usually make reference to the word ``folk'' in homotopic construction 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-prederivator of $(\oo\Cat,\W_{\oo}^{\folk})$ and
\[
......@@ -560,7 +560,7 @@ For later reference, we put here the following trivial but important lemma, whos
It follows from the previous theorem and Theorem \ref{thm:cisinskiI} that the localizer $(\oo\Cat,\W_{\oo}^{\folk})$ is homotopy cocomplete.
\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:
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:
\begin{itemize}[label=-]
\item for every $y \in D_0$, there exists $x \in C_0$ such that
\[
......@@ -578,7 +578,7 @@ For later reference, we put here the following trivial but important lemma, whos
\[
F(\alpha)=\beta.
\]
This characterization of folk trivial fibrations is to be compared with Definition \ref{def:eqomegacat} of equivalences of $\oo$\nbd-categories.
This characterization of folk trivial fibrations is to be compared with Definition \ref{def:eqomegacat} of equivalences of $\oo$\nbd{}categories.
\end{itemize}
\end{paragr}
\begin{proposition}\label{prop:freeiscofibrant}
......@@ -627,11 +627,11 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
Every equivalence of $\oo$-categories is a Thomason equivalence.
\end{proposition}
\begin{remark}
The converse of the above proposition is false. For example, the unique $\oo$\nbd-functor
The converse of the above proposition is false. For example, the unique $\oo$\nbd{}functor
\[
\sD_1 \to \sD_0
\]
is a Thomason equivalence because its image by the nerve is the unique morphism of simplicial sets $\Delta_1 \to \Delta_0$ (which obviously is a weak equivalence), but it is \emph{not} an equivalence of $\oo$\nbd-categories because $\sD_1$ and $\sD_0$ are not equivalent as categories (see Example \ref{example:equivalencecategories}).
is a Thomason equivalence because its image by the nerve is the unique morphism of simplicial sets $\Delta_1 \to \Delta_0$ (which obviously is a weak equivalence), but it is \emph{not} an equivalence of $\oo$\nbd{}categories because $\sD_1$ and $\sD_0$ are not equivalent as categories (see Example \ref{example:equivalencecategories}).
\end{remark}
\begin{paragr}\label{paragr:compweakeq}
Proposition \ref{prop:folkisthom} implies that the identity functor on $\oo\Cat$ induces a morphism of localizers $(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th})$, which in turn induces a functor between localized categories
......@@ -657,11 +657,11 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\ar[from=1-1,to=2-2,phantom,very near start,"\lrcorner"]
\end{tikzcd}
\]
We also define an $\oo$\nbd-functor $\pi : A/a_0 \to A$ as the following composition
We also define an $\oo$\nbd{}functor $\pi : A/a_0 \to A$ as the following composition
\[
\pi : A/a_0 \to \homlax(\sD_1,A) \overset{\pi^A_0}{\longrightarrow} A.
\]
Let us now give an alternative definition of the $\oo$\nbd-category $A/a_0$ using explicit formulas. The equivalence with the previous definition follows from the dual of \cite[Proposition B.5.2]{ara2016joint}
Let us now give an alternative definition of the $\oo$\nbd{}category $A/a_0$ using explicit formulas. The equivalence with the previous definition follows from the dual of \cite[Proposition B.5.2]{ara2016joint}
\begin{itemize}[label=-]
\item An $n$-cell of $A/a_0$ is a matrix \todo{le mot ``matrix'' est-il maladroit ?}
\[
......@@ -752,7 +752,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
$c'_i=a'_i\comp_k b'_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x'_k$&for every $k+1 \leq i \leq n$.\\
\end{tabular}
\end{itemize}
We leave it to the reader to check that the formulas are well defined and that the axioms of $\oo$-category are satisfied. The canonical forgetful $\oo$\nbd-functor $\pi : A/a_0 \to A$ is simply expressed as:
We leave it to the reader to check that the formulas are well defined and that the axioms of $\oo$-category are satisfied. The canonical forgetful $\oo$\nbd{}functor $\pi : A/a_0 \to A$ is simply expressed as:
\begin{align*}
A/a_0 &\to A \\
(x,a) &\mapsto x_n.
......@@ -763,7 +763,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\]
\end{paragr}
\begin{example}\label{example:slicecategories}
For a small category $A$ (considered as an $\oo$\nbd-category) and an object $a_0$ of $A$, the category $A/a_0$ in the sense of the previous paragraph is nothing but the usual slice category of $A$ over $a_0$.
For a small category $A$ (considered as an $\oo$\nbd{}category) and an object $a_0$ of $A$, the category $A/a_0$ in the sense of the previous paragraph is nothing but the usual slice category of $A$ over $a_0$.
\end{example}
\begin{paragr}
Let $u : A \to B$ be a morphism of $\oo\Cat$ and $b_0$ an object of $B$. We define the $\oo$-category $A/b_0$ (also denoted $u\downarrow b_0$) as the following fibred product:
......@@ -796,12 +796,12 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\]
is an $n$-cell of $B/b_0$.
The canonical $\oo$\nbd-functor $A/b_0 \to A$ is simply expressed as
The canonical $\oo$\nbd{}functor $A/b_0 \to A$ is simply expressed as
\begin{align*}
A/b_0 &\to A\\
(x,b) &\mapsto x_n,
\end{align*}
and the $\oo$\nbd-functor $u/b_0$ as
and the $\oo$\nbd{}functor $u/b_0$ as
\begin{align*}
u/b_0 : A/b_0 &\to B/b_0 \\
(x,b) &\mapsto (u(x),b).
......@@ -834,11 +834,11 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
is an equivalence of $\oo$-categories, then so is $u$.
\end{proposition}
\begin{proof}
Before anything else, let us note the following trivial but important fact: for any $\oo$\nbd-functor $F : X \to Y$ and any $n$-cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$.
Before anything else, let us note the following trivial but important fact: for any $\oo$\nbd{}functor $F : X \to Y$ and any $n$-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 $0$\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 $f$ 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 $u(\alpha) \sim_{\oo} \beta$.
\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 $u(\alpha) \sim_{\oo} \beta$.
Let us use the notations:
\begin{itemize}[label=-]
......@@ -866,7 +866,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
& (a_n',1_{v(a_n')})
\end{pmatrix}
\]
are parallel $n$\nbd-cells of $A/c_0$ where we set $c_0:=v(a_0')$. Similarly, we have an $(n+1)$\nbd-cell of $B/c_0$
are parallel $n$\nbd{}cells of $A/c_0$ where we set $c_0:=v(a_0')$. Similarly, we have an $(n+1)$\nbd{}cell of $B/c_0$
\[
\begin{pmatrix}
\begin{matrix}
......@@ -876,7 +876,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
& (\beta,1_{w(\beta)})
\end{pmatrix}
\]
whose source and target respectively are the image by $u/c_0$ of the above two cells of $A/c_0$. By hypothesis, there exists an $(n+1)$\nbd-cell of $A/c_0$ of the form
whose source and target respectively are the image by $u/c_0$ of the above two cells of $A/c_0$. By hypothesis, there exists an $(n+1)$\nbd{}cell of $A/c_0$ of the form
\[
\begin{pmatrix}
\begin{matrix}
......@@ -886,12 +886,12 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
& (\alpha,\Lambda)
\end{pmatrix}
\]
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}
\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.}
\begin{paragr} The name ``folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd-categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one.
\begin{paragr} The name ``folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one.
\end{paragr}
\begin{proposition}[Ara and Maltsiniotis' Theorem A] Let
\[
......@@ -907,3 +907,8 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
is a Thomason equivalence, then so is $u$.
\end{proposition}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:
......@@ -43,7 +43,7 @@ For later reference, we put here the following definition.
the morphism $f'$ is also a weak equivalence.
\end{definition}
\begin{paragr}
A \emph{morphism of localizers} $F : (\C,\W) \to (\C',\W')$ is a functor $F~:~\C~\to~\C'$ that preserves weak equivalences, i.e. such that $F(\W) \subseteq \W'$. The universal property of the localization implies that $F$ induces a canonical functor
A \emph{morphism of localizers} $F : (\C,\W) \to (\C',\W')$ is a functor $F:\C\to\C'$ that preserves weak equivalences, i.e. such that $F(\W) \subseteq \W'$. The universal property of the localization implies that $F$ induces a canonical functor
\[
\overline{F} : \ho(\C) \to \ho(\C')
\]
......@@ -644,7 +644,7 @@ We now turn to the most important way of obtaining op-prederivators.
\section{Model categories}
In this section, we quickly review some aspects of relation between Quillen's theory of model categories and Grothendieck's theory of derivators. We suppose that the reader is familiar with the former one and refer to the standard textbooks on the subject (such as \cite{hovey2007model}, \cite{hirschhorn2009model} or \cite{dwyer1995homotopy}) for basic definitions and results.
For a model category $\M = (\M,\W,\Cof,\Fib)$, the homotopy op-prederivator of $\M$, denoted by $\Ho(\M)$, is the homotopy op-prederivator of the localizer $(\M,\W)$.
For a model category $\M = (\M,\W,\Cof,\Fib)$, the homotopy op\nbd{}prederivator of $\M$, denoted by $\Ho(\M)$, is the homotopy op-prederivator of the localizer $(\M,\W)$.
The following theorems are due to Cisinski \cite{cisinski2003images} and can be summed up by the slogan:
\begin{center}
......@@ -766,3 +766,8 @@ By definition of the projective model structure, $u^*$ preserve weak equivalence
\begin{proof}
See for example \cite[Proposition A.2.4.4(i)]{lurie2009higher}.
\end{proof}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:
\chapter*{Introduction}
The general framework in which the work to be presented in this dissertation takes place is the \emph{homotopy theory of strict $\oo$-categories}, and, as the title suggests, the focus is on homological aspects of this theory. The goal pursued is to study and compare two different homological invariants for strict $\oo$\nbd-categories; that is to say, two different functors \[\mathbf{Str}\oo\Cat \to \ho(\Ch)\] from the category of strict $\oo$-categories to the homotopy category of chain complexes in non-negative degree (i.e.\ the localization of the category of chain complexes in non-negative degree with respect to quasi-isomorphisms). Before entering the heart of the subject and explaining precisely what the above means, let us immediately mention that, with the only exception of the end of this introduction, all the $\oo$-categories considered will be strict. Hence, we drop the adjective ``strict'' and simply say \emph{$\oo$-category} instead of \emph{strict $\oo$-category}. Consequently, we write $\oo\Cat$ instead of $\mathbf{Str}\oo\Cat$ for the category of (strict) $\oo$-categories.
The general framework in which the work to be presented in this dissertation takes place is the \emph{homotopy theory of strict $\oo$-categories}, and, as the title suggests, the focus is on homological aspects of this theory. The goal pursued is to study and compare two different homological invariants for strict $\oo$\nbd{}categories; that is to say, two different functors \[\mathbf{Str}\oo\Cat \to \ho(\Ch)\] from the category of strict $\oo$-categories to the homotopy category of chain complexes in non-negative degree (i.e.\ the localization of the category of chain complexes in non-negative degree with respect to quasi-isomorphisms). Before entering the heart of the subject and explaining precisely what the above means, let us immediately mention that, with the only exception of the end of this introduction, all the $\oo$-categories considered will be strict. Hence, we drop the adjective ``strict'' and simply say \emph{$\oo$-category} instead of \emph{strict $\oo$-category}. Consequently, we write $\oo\Cat$ instead of $\mathbf{Str}\oo\Cat$ for the category of (strict) $\oo$-categories.
\begin{named}[Background: $\oo$-categories as spaces] The homotopy theory of $\oo$-categories most certainly started with the introduction by Street \cite{street1987algebra} of a nerve functor
\begin{named}[Background: $\oo$-categories as spaces] The homotopy theory of $\oo$\nbd{}categories most certainly started with the introduction by Street \cite{street1987algebra} of a nerve functor
\[
N_{\omega} : \oo\Cat \to \Psh{\Delta}
\]
......@@ -10,7 +10,7 @@ that associates to any $\oo$-category $C$ a simplicial set $N_{\oo}(C)$ called t
\[
\overline{N_{\omega}} : \ho(\oo\Cat^{\Th}) \to \ho(\Psh{\Delta}),
\]
where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thomason weak equivalences and $\ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with respect to (Quillen) weak equivalences. As it happens, the functor $\overline{N_{\omega}}$ is an equivalence of categories, as proved by Gagna in \cite{gagna2018strict}. In other words, the homotopy theory of $\oo$-categories induced by Thomason weak equivalences is the same as the homotopy theory of spaces. Gagna's result is in fact a generalization of the analoguous result for the usual nerve of small categories, which is attributed to Quillen in \cite{illusie1972complexe}. In the case of small categories, Thomason even showed the existence of a model structure whose weak equivalences are the ones induced by the nerve \cite{thomason1980cat}. The analogous result for $\oo\Cat$ is conjectured but not yet established \cite{ara2014vers}.
where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thomason weak equivalences and $\ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with respect to (Quillen) weak equivalences. As it happens, the functor $\overline{N_{\omega}}$ is an equivalence of categories, as proved by Gagna in \cite{gagna2018strict}. In other words, the homotopy theory of $\oo$\nbd{}categories induced by Thomason weak equivalences is the same as the homotopy theory of spaces. Gagna's result is in fact a generalization of the analoguous result for the usual nerve of small categories, which is attributed to Quillen in \cite{illusie1972complexe}. In the case of small categories, Thomason even showed the existence of a model structure whose weak equivalences are the ones induced by the nerve \cite{thomason1980cat}. The analogous result for $\oo\Cat$ is conjectured but not yet established \cite{ara2014vers}.
\end{named}
\begin{named}[Two homologies for $\oo$-categories]
Having in mind the nerve functor of Street, a most natural thing to do is to define the \emph{$k$-th homology group of an $\oo$-category $C$} as the $k$-th homology group of the nerve of $C$. In light of Gagna's result, these homology groups are just another way of looking at the homology groups of spaces. In order to explicitly avoid future confusion, we shall now use the name \emph{Street homology groups} of $C$ for these homology groups and use the notation $H^{\St}_k(C)$.
......@@ -30,7 +30,7 @@ where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thom
\text{Do we have }H_k^{\pol}(C) \simeq H_k^{\St}(C)\text{ for any }\oo\text{-category }C\text{ ? }
\end{equation}
\fi
A partial answer to this question is given by Lafont and Métayer in \cite{lafont2009polygraphic}~: for a monoid $M$ (seen as category and hence as an $\oo$-category), we have $H_k^{\pol}(M) \simeq H_k^{\St}(M)$. In fact, the original motivation for polygraphic homology was the homology of monoids and is part of a program carried out to generalize to higher dimension the results of Squier on rewriting theory of monoids \cite{guiraud2006termination}, \cite{lafont2007algebra}, \cite{guiraud2009higher}, \cite{guiraud2018polygraphs}. However, interestingly enough, the general answer to the above question is \emph{no}. A counterexample was found by Maltsiniotis and Ara. Let $B$ be the commutative monoid $(\mathbb{N},+)$, seen as a $2$-category with only one $0$-cell and no non-trivial $1$-cell. This $2$-category is free (as an $\oo$\nbd-category) and a quick computation shows that:
A partial answer to this question is given by Lafont and Métayer in \cite{lafont2009polygraphic}~: for a monoid $M$ (seen as category and hence as an $\oo$-category), we have $H_k^{