@@ -179,7 +179,7 @@ From the previous proposition, we deduce the following very useful corollary.

L(C)\ar[r,"L(\gamma)"]& L(D)

\end{tikzcd}

\]

is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.

is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason equivalences.

\end{corollary}

\begin{proof}

Since the nerve $N$ induces an equivalence of op-prederivators

...

...

@@ -229,7 +229,7 @@ From the previous proposition, we deduce the following very useful corollary.

L(C)\ar[r,"L(\gamma)"]&L(D)

\end{tikzcd}

\]

is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.

is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason equivalences.

\end{proposition}

\begin{proof}

The case where $\alpha$ or $\beta$ is both injective on objects and quasi-injective on arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is quasi-injective on arrows; the remaining case being symmetric.

...

...

@@ -295,7 +295,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

\sD_1\ar[r]& C',

\end{tikzcd}

\]

where the morphism $\sS_1\to\sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1\to\sD_1$ is injective on objects and the morphism $\sS_1\to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}.

where the morphism $\sS_1\to\sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square is homotopy cocartesian in $\Cat$ (when equipped with Thomason equivalences). Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1\to\sD_1$ is injective on objects and the morphism $\sS_1\to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}.

\end{example}

\begin{example}[Killing a generator]

Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by ``killing'' $f$, i.e. defined with the following cocartesian square:

...

...

@@ -305,7 +305,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

\sD_0\ar[r]& C'.

\end{tikzcd}

\]

Then, this above square is homotopy cocartesion in $\Cat$ (equipped with the Thomason weak equivalences). Indeed, it obviously is the image of a square in $\Rgrph$ by the functor $L$ and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$.

Then, this above square is homotopy cocartesian in $\Cat$ (equipped with Thomason equivalences). Indeed, it obviously is the image of a square in $\Rgrph$ by the functor $L$ and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$.

\end{example}

\begin{remark}

Note that in the previous example, we see that it was useful to consider the category of reflexive graphs and not only the category of graphs because the map $\sD_1\to\sD_0$ does not come from a morphism in the category of graphs. \todo{À mieux dire ?}

...

...

@@ -579,7 +579,7 @@ In the definition of the bisimplicial nerve of a $2$-category we gave, we have p

which enables us to compare the homotopy theory of $2\Cat$ with the homotopy theory of bisimplicial sets.

\end{paragr}

\begin{lemma}\label{lemma:binervthom}

A $2$-functor $F : C \to D$ is a Thomason weak equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets.

A $2$-functor $F : C \to D$ is a Thomason equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets.

\end{lemma}

\begin{proof}

It follows from what is shown in \cite[Section 2.1 and Theorem 2.7]{bullejos2003geometry} that there is weak equivalence of simplicial sets

...

...

@@ -588,7 +588,7 @@ In the definition of the bisimplicial nerve of a $2$-category we gave, we have p

\]

which is natural in $C$. This implies what we wanted to show.

\end{proof}

From this lemma, we deduce two useful criteria to detect Thomason weak equivalences of $2$-categories.

From this lemma, we deduce two useful criteria to detect Thomason equivalences of $2$-categories.

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

Let $F : C \to D$ be a $2$-functor. If

\begin{enumerate}[label=\alph*)]

...

...

@@ -600,9 +600,9 @@ From this lemma, we deduce two useful criteria to detect Thomason weak equivalen

\[

C(x,y)\to D(F(x),F(y))

\]

induced by $F$ is a Thomason weak equivalence of $1$-categories,

induced by $F$ is a Thomason equivalence of $1$-categories,

\end{enumerate}

then $F$ is a Thomason weak equivalence of $2$-categories.

then $F$ is a Thomason equivalence of $2$-categories.

\end{corollary}

\begin{proof}

By definition, for every $2$-category $C$ and every $m \geq0$, we have

...

...

@@ -613,7 +613,7 @@ From this lemma, we deduce two useful criteria to detect Thomason weak equivalen

\end{proof}

\begin{corollary}

Let $F : C \to D$ be a $2$-functor. If for every $k \geq0$,

\[V(F)_k : V(C)_k \to V(D)_k\] is a Thomason weak equivalence of $1$-categories, then $F$ is a Thomason weak equivalence of $2$-categories.

\[V(F)_k : V(C)_k \to V(D)_k\] is a Thomason equivalence of $1$-categories, then $F$ is a Thomason equivalence of $2$-categories.

\end{corollary}

\begin{proof}

From Lemma \ref{lemma:binervehorizontal}, we now that for every $m \geq0$,

...

...

@@ -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 weak equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)}\to A_{(1,n)}$ is also a (co-universal) Thomason weak 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 $m \geq0$, by considering the cocartesian square

\[

...

...

@@ -763,7 +763,7 @@ For any $n \geq 0$, consider the following cocartesian square

A_{(m,1)}\ar[r]& A_{(m,n)},

\end{tikzcd}

\]

where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This $2$-functor is once again a folk cofibration, but it is \emph{not} in general a co-universal Thomason weak equivalence (it is if we make the hypothesis that $m\neq0$, but we did not). However, since we made the hypothesis that $n\neq0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta\to\Delta_n$ is a co-universal Thomason weak equivalence. Hence, the previous square is Thomason homotopy cocartesian and $A_{(m,n)}$ has the homotopy type of a point. Since $A_{(m,1)}$, $\Delta_1$ and $\Delta_n$ are \good{}, this shows that for $m \geq0$ and $n >0$, $A_{(m,n)}$ is \good{}.

where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This $2$-functor is once again a folk cofibration, but it is \emph{not} in general a co-universal Thomason equivalence (it is if we make the hypothesis that $m\neq0$, but we did not). However, since we made the hypothesis that $n\neq0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta\to\Delta_n$ is a co-universal Thomason equivalence. Hence, the previous square is Thomason homotopy cocartesian and $A_{(m,n)}$ has the homotopy type of a point. Since $A_{(m,1)}$, $\Delta_1$ and $\Delta_n$ are \good{}, this shows that for $m \geq0$ and $n >0$, $A_{(m,n)}$ is \good{}.

Similarly, if $m >0$ and $ n\geq0$, then $A_{(m,n)}$ has the homotopy type of a point and is \good{}.

Every 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}

\]

...

...

@@ -61,17 +61,17 @@ Consider the commutative square

\sH^{\pol}(\sD_0)\ar[r,"\pi_{\sS_0}"]&\sH(\sD_0).

\end{tikzcd}

\]

It follows respectively from Proposition \ref{prop:oplaxhmptyisthom} 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.

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 expected ``lax'' variation and Proposition \ref{prop:contractibleisgood} is still 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}).

\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 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}

\todo{À écrire.}

This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}.

@@ -1151,10 +1151,10 @@ As an immediate consequence of the previous lemma, the functor $\lambda_{\leq n}

\ho(\oo\Cat^{\folk})\ar[r,"\LL\lambda"]&\ho(\Ch)

\end{tikzcd}

\]

is \emph{not} commutative. If it were, then for every $n$\nbd-category $C$ and every $k >n$, we would have $H_k^{pol}(\iota_n(C))=0$ for every $k >n$, which is not even true for the case $n=1$ in the following chapter.

is \emph{not} commutative. If it were, then for every $n$\nbd-category $C$ and every $k >n$, we would have $H_k^{\pol}(\iota_n(C))=0$ for every $k >n$, which is not even true for the case $n=1$ in the following chapter.

\end{remark}

A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the following corollary.

\begin{corollary}

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

Let $n \geq0$ and $C$ be an $\oo$\nbd-category. If $C$ has a $k$\nbd-basis for every $0\leq k \leq n-1$, then the canonical map of $\ho(\Ch)$

\[

\alpha^{\pol}_C : \sH^{\pol}(C)\to\lambda(C)

...

...

@@ -1168,7 +1168,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi

\begin{proof}

\todo{À écrire}

\end{proof}

\begin{paragr}

\begin{paragr}\label{paragr:polhmlgylowdimension}

Since every $\oo$\nbd-category trivially admits its set of $0$\nbd-cells as a $0$\nbd-base, it follows from the previous proposition that for every $\oo$\nbd-category $C$ we have

\[

\sH^{\pol}_0(C)\simeq H_0(\lambda(C))

...

...

@@ -1264,7 +1264,7 @@ Straighforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and th

and a thorough analysis of naturality shows that this isomorphism is nothing but the canonical morphism $c_1N_{\oo}(C)\to\tau_{\leq1}^{i}(C)$.

\end{proof}

We can now prove the important following proposition.

From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha^{\sing}$ induces isomorphisms $H_k^{\sing}(C)\simeq H_k(\lambda(C))$ for $k \in\{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha^{\pol}$ induces isomorphisms $H_k^{\pol}(C)\simeq H_k(\lambda(C))$ for $k \in\{0,1\}$. The result follows then from an immediate 2-out-of-3 property.

\end{proof}

\begin{paragr}

A natural question following the above proposition is:

\begin{center}

For which $k \geq0$ do we have $H_k^{\sing}(C)\simeq H_k^{\pol}(C)$ for every $\oo$\nbd-category $C$ ?

\end{center}

We have already seen in \ref{paragr:bubble} that when $C = B^2\mathbb{N}$ we have

for every $p \geq2$. Furthermore, with a similar argument to the one given in \ref{paragr:bubble}, we have that for every $k \geq3$, the (nerve of the) $\oo$\nbd-category $B^k\mathbb{N}$ is a $K(\mathbb{Z},k)$-space. In particular, we have

for every $p \geq1$ (see \cite[Theorem 23.1]{eilenberg1954groups}). On the other hand, since $B^k\mathbb{N}$ is a free $k$\nbd-category, we have $H_n^{\pol}(B^k\mathbb{N})=0$ for all $n \geq k$. All in all, we have proved that for every $k \geq4$, there exists at least one $\oo$\nbd-category $C$ such that

\[

H_k^{\sing}(C)\not\simeq H_k^{\pol}(C).

\]

However, it is still an open question to know whether for $k \in\{2,3\}$ we have

\[

H^{\sing}_k(C)\simeq H^{\pol}_k(C).

\]

The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of $k$ is the analoguous of Lemma \ref{lemma:truncationcounit}. But contrary to the case $k=1$, it is not generally true that the canonical morphism $c_k N_{\oo}(C)\to\tau^{i}_{\leq k}(C)$ is an isomorphism when $k \geq2$. However, what we really need is that the image by $\lambda$ of this morphism be a quasi-isomorphism. In the case $k=2$, it seems that this canonical morphism admits an oplax $2$\nbd-functor as an inverse up to oplax transformation which could be an hint towards the conjecture that $H^{\sing}_2(C)\simeq H^{\pol}_2(C)$ for every $\oo$\nbd-category $C$.

\end{paragr}

%% Slightly less trivial is the following lemma.

%% \begin{lemma}

%% The following triangle of functors

...

...

@@ -1315,7 +1348,3 @@ Finally, we obtain the result we were aiming for.

%% \end{lemma}

%%\section{Homology and Homotopy of $\oo$-categories in low dimension}

\abstract{In this dissertation, we study the homology of strict $\oo$-categories. More precisely, we intend to compare the ``classical'' homology of an $\oo$-category (defined as the homology of its Street nerve) with its polygraphic homology. Along the way, we prove several important result concerning free strict $\oo$\nbd-categories over polygraphs-or-computad and concerning the homotopy theory of strict $\oo$\nbd-categories. }