@@ -27,7 +27,7 @@ We denote by $\Ch$ the category of non-negatively graded chain complexes of abel

\[

\kappa_n(X)=K_n(X)/D_n(X).

\]

Using the simplicial identities, it can be shown that $\partial(D_n(X))\subseteq D_{n-1}(X)$ for every $n>0$. Hence, an induced differential that we still denote by $\partial$:

Using the simplicial identities, it can be shown that $\partial(D_n(X))\subseteq D_{n-1}(X)$ for every $n>0$. Hence, there is an induced differential which we still denote by $\partial$:

\[

\partial : \kappa_n(X)\to\kappa_{n-1}(X).

\]

...

...

@@ -63,7 +63,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

where we wrote $\W_{\Ch}$ for the class of quasi-isomorphisms.

\end{paragr}

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

The \emph{singular homology functor for $\oo$-categories}$\sH^{\sing}$ is defined as the following composition

The \emph{singular homology functor for $\oo$\nbd{}categories}$\sH^{\sing}$ is defined as the following composition

@@ -78,7 +78,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

\end{paragr}

%% \begin{paragr}

%% In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.

%% In simpler words, the homology of an $\oo$\nbd{}category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.

%% Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.

%% \end{paragr}

...

...

@@ -86,7 +86,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

The adjective ``singular'' is there to avoid future confusion with another

homological invariant for $\oo$\nbd{}categories that will be introduced later.

As a matter of fact, the underlying point of view adopted in this thesis is

that \emph{singular homology of $\oo$-categories} ought to be simply called

that \emph{singular homology of $\oo$\nbd{}categories} ought to be simply called

\emph{homology of $\oo$\nbd{}categories} as it is the only ``correct''

definition of homology. This assertion will be justified in Remark \ref{remark:polhmlgyisnotinvariant}.

\end{remark}

...

...

@@ -100,19 +100,19 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

The singular homology $\sH^{\sing} : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch)$ is homotopy cocontinuous.

The singular homology \[\sH^{\sing} : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch)\] is homotopy cocontinuous.

\end{proposition}

\begin{proof}

This follows from the fact that $\overline{N_{\oo}}$ and $\overline{\kappa}$ are both homotopy cocontinuous. In the first case, this is because $\overline{N_{\oo}}$ is an equivalence of op-prederivators and thus we can apply Lemma \ref{lemma:eqisadj} and Lemma \ref{lemma:ladjcocontinuous}. In the second case, this is because $\kappa$ is left Quillen and we can apply Theorem \ref{thm:cisinskiII}.

This follows from the fact that $\overline{N_{\oo}}$ and $\overline{\kappa}$ are both homotopy cocontinuous. For $\overline{N_{\oo}}$, this is because it is an equivalence of op\nbd{}prederivators and thus we can apply Lemma \ref{lemma:eqisadj} and Lemma \ref{lemma:ladjcocontinuous}. For $\overline{\kappa}$, this is because $\kappa$ is left Quillen and thus we can apply Theorem \ref{thm:cisinskiII}.

\end{proof}

\section{Abelianization}

We write $\Ab$ for category of abelian groups and for an abelian group $G$, we write $\vert G \vert$ for the underlying set of $G$.

\begin{paragr}

Let $C$ be an $\oo$-category. For every $n\geq0$, we define $\lambda_n(C)$ as the abelian group obtained by quotienting $\mathbb{Z}C_n$, the free abelian group on $C_n$, by the congruence generated by the relations

Let $C$ be an $\oo$\nbd{}category. For every $n\geq0$, we define $\lambda_n(C)$ as the abelian group obtained by quotienting $\mathbb{Z}C_n$(the free abelian group on $C_n$) by the congruence generated by the relations

\[

x \comp_k y \sim x+y

\]

for all $x,y \in C_n$ that are $k$-composable for some $k<n$. For $n=0$, this means that $\lambda_0(C)=\mathbb{Z}C_0$. Now let $f : C \to D$ be an $\oo$-functor. For every $n \geq0$, the map

for all $x,y \in C_n$ that are $k$-composable for some $k<n$. For $n=0$, this means that $\lambda_0(C)=\mathbb{Z}C_0$. Now let $f : C \to D$ be an $\oo$\nbd{}functor. For every $n \geq0$, the definition of $\oo$\nbd{}functors implies that the map

\begin{align*}

\mathbb{Z}C_n &\to\mathbb{Z}D_{n}\\

x \in C_n &\mapsto f(x)

...

...

@@ -121,14 +121,14 @@ We write $\Ab$ for category of abelian groups and for an abelian group $G$, we w

\[

\lambda_n(f) : \lambda_n(C)\to\lambda_n(D).

\]

and this obviously defines a functor $\lambda_n : \oo\Cat\to\Ab$.

This defines a functor $\lambda_n : \oo\Cat\to\Ab$.

For $n>0$, consider the linear map

\begin{align*}

\mathbb{Z}C_n &\to\mathbb{Z}C_{n-1}\\

x \in C_n &\mapsto t(x)-s(x).

\end{align*}

The axioms of $\oo$-categories imply that it induces a map

The axioms of $\oo$\nbd{}categories imply that it induces a map

\[

\partial : \lambda_{n}(C)\to\lambda_{n-1}(C)

\]

...

...

@@ -161,7 +161,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

\end{align*}

where $\Ab$ is the category of abelian groups.

On the other hand, write again $\lambda_n$ for the functor

Besides, let us write $\lambda_n$again for the functor

\begin{align*}

\lambda_n : n\Cat&\to\Ab\\

C&\mapsto\lambda_n(C).

...

...

@@ -174,7 +174,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

\begin{proof}

The case $n=0$ is immediate since the functor $\lambda_0 : 0\Cat=\Set\to\Ab$ is the ``free abelian group'' functor and the functor $B^0 : \Ab\to0\Cat=\Set$ is the ``underlying set'' functor.

Suppose now that $n >0$. From Lemma \ref{lemma:nfunctortomonoid} we know that for any abelian group $G$ and any $n$\nbd{}category $C$, the map

Suppose now that $n >0$. From Lemma \ref{lemma:nfunctortomonoid} we know that for every abelian group $G$ and every $n$\nbd{}category $C$, the map

\begin{align*}

\Hom_{n\Cat}(C,B^nG) &\to\Hom_{\Set}(C_n,\vert G \vert)\\

F &\mapsto F_n,

...

...

@@ -198,7 +198,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

\[

f(1_x)=0

\]

because any element of an abelian group has an inverse. Now, because of the adjunction morphism

because every element of an (abelian) group has an inverse. Now, because of the adjunction morphism

\[

\Hom_{\Set}(C_n,\vert G \vert)\simeq\Hom_{\Ab}(\mathbb{Z}C_n,G),

\]

...

...

@@ -212,14 +212,14 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

\]

\end{proof}

\begin{paragr}\label{paragr:abelpolmap}

Let $C$ be an $\oo$-category, $n \in\mathbb{N}$ and $E \subseteq C_n$ a subset of the $n$-cells. We obtain a map $\mathbb{Z}E \to\lambda_n(C)$ defined as the composition

Let $C$ be an $\oo$\nbd{}category, $n \in\mathbb{N}$ and $E \subseteq C_n$ a subset of the $n$-cells. We obtain a map $\mathbb{Z}E \to\lambda_n(C)$ defined as the composition

\[

\mathbb{Z}E \to\mathbb{Z}C_n \to\lambda_n(C),

\]

where the map on the left is induced by the canonical inclusion of $E$ in $C_n$ and the map on the right is the quotient map from the definition of $\lambda_n(C)$.

\end{paragr}

\begin{lemma}\label{lemma:abelpol}

Let $C$ be a \emph{free}$\oo$-category and let $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$ be its basis. For every $n \in\mathbb{N}$, the map

Let $C$ be a \emph{free}$\oo$\nbd{}category and let $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$ be its basis. For every $n \in\mathbb{N}$, the map

\[

\mathbb{Z}\Sigma_n \to\lambda_n(C)

\]

...

...

@@ -250,7 +250,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

%% \end{aligned}

%% \]

%%

Notice first that for any $\oo$\nbd{}category $C$, we have $\lambda_n(\tau_{\leq n}^s(C))=\lambda_n(C)$. Suppose now that $C$ is free with basis $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$. Using Lemma \ref{lemma:adjlambdasusp} and Lemma \ref{lemma:freencattomonoid}, we obtain that for any abelian group $G$, we have

Notice first that for every $\oo$\nbd{}category $C$, we have $\lambda_n(\tau_{\leq n}^s(C))=\lambda_n(C)$. Suppose now that $C$ is free with basis $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$. Using Lemma \ref{lemma:adjlambdasusp} and Lemma \ref{lemma:freencattomonoid}, we obtain that for every abelian group $G$, we have

@@ -259,7 +259,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c

\end{align*}

\end{proof}

\begin{paragr}

Let $C$ be a \emph{free}$\oo$-category and write $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$ for its basis. For every $n \geq0$ and every $\alpha\in\Sigma_n$, recall that we have proved in Proposition \ref{prop:countingfunction} the existence of a unique function $w_{\alpha} : C_n \to\mathbb{N}$ such that:

Let $C$ be a \emph{free}$\oo$\nbd{}category and write $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$ for its basis. For every $n \geq0$ and every $\alpha\in\Sigma_n$, recall that we have proved in Proposition \ref{prop:countingfunction} the existence of a unique function $w_{\alpha} : C_n \to\mathbb{N}$ such that:

In the next Lemma, recall the definition of \emph{homotopy of chain complexes}(for example from \cite[Definition 1.4.4]{weibel1995introduction} where it is called \emph{chain homotopy}).

For the definition of \emph{homotopy of chain complexes}see for example \cite[Definition 1.4.4]{weibel1995introduction}(where it is called \emph{chain homotopy}).

\begin{lemma}\label{lemma:abeloplax}

Let $u, v : C \to D$ be two $\oo$-functors. If there is an oplax transformation $\alpha : u \Rightarrow v$, then there is a homotopy of chain complexes from $\lambda(u)$ to $\lambda(v)$.

Let $u, v : C \to D$ be two $\oo$\nbd{}functors. If there is an oplax transformation $\alpha : u \Rightarrow v$, then there is a homotopy of chain complexes from $\lambda(u)$ to $\lambda(v)$.

\end{lemma}

\begin{proof}

For any$n$-cell $x$ of $C$ (resp.\ $D$), let us use the notation $[x]$ for the image of $x$ in $\lambda_n(C)$ (resp.\ $\lambda_n(D)$).

For an $n$-cell $x$ of $C$ (resp.\ $D$), let us use the notation $[x]$ for the image of $x$ in $\lambda_n(C)$ (resp.\ $\lambda_n(D)$).

Let $h_n$ be the map

\[

...

...

@@ -398,7 +398,7 @@ In the next Lemma, recall the definition of \emph{homotopy of chain complexes} (

\]

be commutative squares in $\omega\Cat$ for $\epsilon\in\{0,1\}$.

If $C'$ is a free $\omega$-category and $v$ folk trivial fibration, then for any oplax transformation \[\alpha : f_0\Rightarrow f_1,\] there is an oplax transformation \[\alpha' : f_0' \Rightarrow f_1'\] such that

If $C'$ is a free $\omega$-category and $v$ folk trivial fibration, then for any oplax transformation \[\alpha : f_0\Rightarrow f_1,\] there exists an oplax transformation \[\alpha' : f_0' \Rightarrow f_1'\] such that

\[

v \star\alpha' =\alpha\star u.

\]

...

...

@@ -413,7 +413,7 @@ In the next Lemma, recall the definition of \emph{homotopy of chain complexes} (

\]

(where $i_1 : \sD_0\amalg\sD_0\simeq\sS_0\to\sD_1$ is the morphism introduced in \ref{paragr:inclusionsphereglobe}) is commutative.

Now, the hypothesis of the lemma yield the following commutative square

Now, the hypotheses of the lemma yield the following commutative square

\[

\begin{tikzcd}

(\sD_0\amalg\sD_0)\otimes C' \ar[d,"{i_1\otimes C'}"']\ar[rr,"{\langle f'_0, f_1' \rangle}"]&& D' \ar[d,"v"]\\

...

...

@@ -426,7 +426,7 @@ In the next Lemma, recall the definition of \emph{homotopy of chain complexes} (

\]

The commutativity of the two induced triangles shows what we needed to prove.

\end{proof}

From now on, when given an $\oo$\nbd{}functor $u$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma_{\folk}(u))$ (where $\gamma_{\folk}$ is the localization functor $\oo\Cat\to\ho(\oo\Cat^{\folk})$) for the morphism induced by $u$ at the level of polygraphic homology.

From now on, when given an $\oo$\nbd{}functor $u$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$ (where $\gamma^{\folk}$ is the localization functor $\oo\Cat\to\ho(\oo\Cat^{\folk})$) for the morphism induced by $u$ at the level of polygraphic homology.

\begin{lemma}

Let $u,v : C \to D$ two $\oo$\nbd{}functors. If there exists an oplax transformation $u\Rightarrow v$, then

\[

...

...

@@ -469,7 +469,7 @@ The following proposition is an immediate consequence of the previous lemma.

is an isomorphism.

\end{proposition}

\begin{paragr}\label{paragr:polhmlgythomeq}

Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder whether it is true that \emph{any} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later, it is not the case.

Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder whether it is true that \emph{any} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later (Proposition \ref{prop:polhmlgynotinvariant}), it is not the case.

\end{paragr}

\begin{remark}

Proposition \ref{prop:oplaxhmtpypolhmlgy} is also true if we replace ``oplax'' by ``lax''.

...

...

@@ -503,12 +503,12 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

Let $\nu : \Ch\to\oo\Cat$ be the right adjoint to the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason equivalences.

\end{lemma}

\begin{proof}

We have already seen that $\lambda : \oo\Cat\to\Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$. By adjunction, this means that $\nu$ is right Quillen for this model structure. In particular, it sends trivial fibrations of chain complexes to folk trivial fibrations. From Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} and the fact that all chain complexes are fibrant, it follows that $\nu$ sends weak equivalences of chain complexes to weak equivalences of the folk model structure, which are in particular Thomason equivalences (see Lemma \ref{lemma:nervehomotopical}).

We have already seen that $\lambda : \oo\Cat\to\Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$. By adjunction, this means that $\nu$ is right Quillen for this model structure. In particular, it sends trivial fibrations of chain complexes to folk trivial fibrations. From Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} and the fact that all chain complexes are fibrant, it follows that $\nu$ sends weak equivalences of chain complexes to weak equivalences of the folk model structure, which are in particular Thomason equivalences (Lemma \ref{lemma:nervehomotopical}).

\end{proof}

\begin{remark}

The proof of the previous lemma shows the stronger result that $\nu$ sends weak equivalences of chain complexes to weak equivalences for the folk model structure on $\oo\Cat$. This will be of no use in the sequel.

\end{remark}

Recall that $c_{\oo} : \Psh{\Delta}\to\oo\Cat$is the left adjoint of the nerve functor ${N_{\oo} : \oo\Cat\to\Psh{\Delta}}$ (see Paragraph \ref{paragr:nerve}).

Recall that we write $c_{\oo} : \Psh{\Delta}\to\oo\Cat$for the left adjoint of the nerve functor ${N_{\oo} : \oo\Cat\to\Psh{\Delta}}$ (see Paragraph \ref{paragr:nerve}).

\begin{lemma}\label{lemma:abelor}

The triangle of functors

\[

...

...

@@ -528,11 +528,11 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

\end{lemma}

We can now state and prove the promised result.

\begin{theorem}\label{thm:hmlgyderived}

Consider that $\oo\Cat$ is equipped with the Thomason equivalences. The abelianization functor $\lambda : \oo\Cat\to\Ch$ is strongly left derivable and the left derived morphism

Consider that $\oo\Cat$ is equipped with the Thomason equivalences. The abelianization functor $\lambda : \oo\Cat\to\Ch$ is strongly left derivable and the left derived morphism of op\nbd{}prederivators

\[

\LL\lambda^{\Th} : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch)

\]

is isomorphic to the Singular homology

is isomorphic to the singular homology

\[

\sH^{\sing} : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch).

\]

...

...

@@ -631,7 +631,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

\[

C:=B^2\mathbb{N}

\]

(see \ref{paragr:suspmonoid}). As usual, we consider $C$ as an $\oo$\nbd{}category with only unit cells above dimension $2$. This $\oo$\nbd{}category is free; namely its $k$\nbd{}basis is a singleton for $k=0$ or $2$ and the empty set otherwise. In particular $C$ is cofibrant for the folk model structure (Proposition \ref{prop:freeiscofibrant}) and it follows from Proposition \ref{prop:abelianizationfreeoocat} that $\sH^{\pol}(C)$ is given by the chain complex (seen as an object of $\ho(\Ch)$)

(see \ref{paragr:suspmonoid}). As usual, we consider $C$ as an $\oo$\nbd{}category with only unit cells strictly above dimension $2$. This $\oo$\nbd{}category is free; namely its $k$\nbd{}basis is a singleton for $k=0$ or $2$ and the empty set otherwise. In particular $C$ is cofibrant for the folk model structure (Proposition \ref{prop:freeiscofibrant}) and it follows from Proposition \ref{prop:abelianizationfreeoocat} that $\sH^{\pol}(C)$ is given by the chain complex (seen as an object of $\ho(\Ch)$)

@@ -652,9 +652,12 @@ Another consequence of the above counter-example is the following result, which

is not an isomorphism of $\ho(\Ch)$.

\end{proposition}

\begin{proof}

Suppose the converse, which is that the functor $\sH^{\pol}\circ\gamma^{\folk} : \oo\Cat\to\ho(\Ch)$ sends Thomason equivalences to isomorphisms of $\ho(\Ch)$.

\]sends Thomason equivalences to isomorphisms of $\ho(\Ch)$.

Notice that, because of the inclusion $\W^{\folk}\subseteq\W^{\Th}_{\oo}$, the category $\ho(\oo\Cat^{\Th})$ may be identified with the localization of $\ho(\oo\Cat^{\folk})$ with respect to $\gamma^{\folk}(\W^{\Th}_{\oo})$ and then the localization functor is nothing but

Because of the inclusion $\W^{\folk}\subseteq\W^{\Th}_{\oo}$, the category $\ho(\oo\Cat^{\Th})$ may be identified with the localization of $\ho(\oo\Cat^{\folk})$ with respect to $\gamma^{\folk}(\W^{\Th}_{\oo})$ and then the localization functor is nothing but

\[

\J : \ho(\oo\Cat^{\folk})\to\ho(\oo\Cat^{\Th}).

\]

...

...

@@ -719,7 +722,7 @@ Another consequence of the above counter-example is the following result, which

It follows from the previous result that if we think of $\oo$\nbd{}categories as a

model for homotopy types (see Theorem \ref{thm:gagna}), then the polygraphic

homology of an $\oo$-category is \emph{not} a well defined invariant. This

homology of an $\oo$\nbd{}category is \emph{not} a well defined invariant. This

justifies what we said in remark \ref{remark:singularhmlgyishmlgy}, which is

that \emph{singular homology} is the only ``correct'' homology of $\oo$\nbd{}categories.

\end{remark}

...

...

@@ -757,14 +760,14 @@ Another consequence of the above counter-example is the following result, which

%% \end{equation}

%% is commutative.

Since $\J$ is nothing but the identity on objects, for any $\oo$\nbd{}category $C$, the natural transformation $\pi$ yields a map

Since $\J$ is nothing but the identity on objects, for every $\oo$\nbd{}category $C$, the natural transformation $\pi$ yields a map

\[

\pi_C : \sH^{\sing}(C)\to\sH^{\pol}(C),

\]

which we shall refer to as the \emph{canonical comparison map.}

\end{paragr}

\begin{remark}

When $C$ is free, it follows from the considerations in \ref{paragr:univmor} that the canonical comparison map $\pi_C$ can be identified with the image by $\alpha^{\Ch}$ of the morphism of $\Ch$

When $C$ is free, it follows from the considerations in \ref{paragr:univmor} that the canonical comparison map $\pi_C$ can be identified with the image by $\gamma^{\Ch}$ of the morphism of $\Ch$

\[

\lambda c_{\oo}N_{\oo}(C)\to\lambda(C)

\]

...

...

@@ -780,10 +783,10 @@ Another consequence of the above counter-example is the following result, which

\end{definition}

\begin{paragr}

The rest of this dissertation is devoted to the study of \good{}$\oo$\nbd{}categories. Examples of such $\oo$\nbd{}categories will be presented later. Note that if we think of $\oo$\nbd{}categories up to Thomason equivalences as spaces (which is an informal way of stating Theorem \ref{thm:gagna}), then it follows from Proposition \ref{prop:polhmlgynotinvariant} that polygraphic homology is not well defined. With this perspective, polygraphic homology can be thought of a way to compute singular homology of \good{}$\oo$\nbd{}categories.

The rest of this dissertation is devoted to the study of \good{}$\oo$\nbd{}categories. Examples of such $\oo$\nbd{}categories will be presented later. Following the perspective of Remark \ref{remark:polhmlgyisnotinvariant}, polygraphic homology can be thought of as a way to compute singular homology of \good{}$\oo$\nbd{}categories.

\end{paragr}

%% \section{A criterion to detect \good{} $\oo$\nbd{}categories}

%% We shall now proceed to give an abstract criterion to find \good{} $\oo$-categories. %In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good{} $\oo$-categories.

%% We shall now proceed to give an abstract criterion to find \good{} $\oo$\nbd{}categories. %In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good{} $\oo$\nbd{}categories.

Similarly to \ref{paragr:cmparisonmap}, the morphism of localizers

...

...

@@ -859,14 +862,14 @@ The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd{}cate

\end{proposition}

\begin{proof}

Notice first that all the constructions from \ref{paragr:defcancompmap} may be reproduced \emph{mutatis mutandis} at the level of op-prederivators. Hence, we obtain a $2$\nbd{}morphism of op-prederivators

Notice first that all the constructions from \ref{paragr:defcancompmap} may be reproduced \emph{mutatis mutandis} at the level of op-prederivators. In particular, we obtain a $2$\nbd{}morphism of op-prederivators

@@ -878,7 +881,7 @@ The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd{}cate

\begin{itemize}[label=-]

\item the top and bottom horizontal arrows are induced by $\pi$,

\item the middle horizontal arrow is induced by $\pi$ and the canonical morphism \[\hocolim_{i \in}^{\Th}(d_i)\to\hocolim_{i \in I}^{\folk}(d_i)\] from \ref{paragr:prelimcriteriongoodcat},

\item the top vertical arrows are the canonical morphisms induced by any morphism of op-prederivators (see \ref{paragr:canmorphismcolimit}),

\item the top vertical arrows are the canonical morphisms induced by every morphism of op-prederivators (see \ref{paragr:canmorphismcolimit}),

\item the bottom vertical arrows are induced by the co-cone \[(\varphi_i : d(i)\to\oo\Cat)_{i \in\Ob(I)}.\]

\end{itemize}

Since $\sH^{\pol}$ and $\sH^{\sing}$ are both homotopy cocontinuous (Proposition \ref{prop:singhmlgycocontinuous} and Proposition \ref{prop:polhmlgycocontinuous}), both top vertical arrows are isomorphisms. Hypotheses $(ii)$ and $(iii)$ imply that the bottom vertical arrows are isomorphisms and hypothesis $(i)$ imply that the top horizontal arrow is an isomorphism. By a 2-out-of-3 property, the bottom horizontal arrow is an isomorphism, which means exactly that $C$ is \good{}.

...

...

@@ -894,11 +897,11 @@ The previous proposition admits the following corollary, which will be of great

\]

be a cocartesian square of $\oo\Cat$ such that:

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

\item the $\oo$-categories $A$,$B$ and $C$ are free and \good{},

\item the $\oo$\nbd{}categories $A$,$B$ and $C$ are free and \good{},

\item at least one of the morphisms $u : A \to B$ or $f : A \to C$ is a folk cofibration,

\item the square is Thomason homotopy cocartesian.

\end{enumerate}

Then, the $\oo$-category $D$ is \good{}.

Then, the $\oo$\nbd{}category $D$ is \good{}.

\end{corollary}

\begin{proof}

The fact that $A$,$B$ and $C$ are free and one of the morphism $u$ or $f$ is a folk cofibration ensure that the square is folk homotopy cocartesian. The conclusion follows then from Proposition \ref{prop:criteriongoodcat}.

...

...

@@ -945,7 +948,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

$\tau^{i}_{\leq n}$ sends folk trivial fibrations of $\oo\Cat$ to weak

equivalences of $n\Cat$ (in virtue of

Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model}).

% Let $C$ be an $\oo$-category and let $\eta_C : C \to \iota_n\tau^{i}_{\leq

% Let $C$ be an $\oo$\nbd{}category and let $\eta_C : C \to \iota_n\tau^{i}_{\leq

% n}(C)$ be the unit morphism of the adjunction $\tau^{i}_{\leq n} \dashv

% \iota_n$. Let us prove that for every pair $(x,y)$ of $k$\nbd{}cells of $C$,

% if $k \leq n$ then $x \simeq_{\oo} y$ (see \ref{paragr:ooequivalence}) if

...

...

@@ -1469,7 +1472,7 @@ Finally, we obtain the result we were aiming for.

%% \]

%% is commutative (up to an isomorphism).

%% \end{lemma}

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

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

which we refer to as the \emph{nerve functor for $\oo$-categories}. Furthermore, for every $n \in\mathbb{N}$, we also define a nerve functor for $n$-categories as the restriction of $N_{\oo}$ to $n\Cat$ (seen as a full subcategory of $\oo\Cat$)

which we refer to as the \emph{nerve functor for $\oo$\nbd{}categories}. Furthermore, for every $n \in\mathbb{N}$, we also define a nerve functor for $n$-categories as the restriction of $N_{\oo}$ to $n\Cat$ (seen as a full subcategory of $\oo\Cat$)

Let $X$ be a simplicial set. The $\oo$-category $c_{\oo}(X)$ is free and the set of generating $k$-cells of $c_{\oo}(X)$ is canonically isomorphic the to set of non-degenerate $k$-simplices of $X$.

Let $X$ be a simplicial set. The $\oo$\nbd{}category $c_{\oo}(X)$ is free and the set of generating $k$-cells of $c_{\oo}(X)$ is canonically isomorphic the to set of non-degenerate $k$-simplices of $X$.

\end{lemma}

\fi

\begin{paragr}

...

...

@@ -172,27 +172,27 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

Let $n \in\nbar$. A morphism $f : X \to Y$ of $n\Cat$ is a \emph{Thomason equivalence} when ${N_n(f) : N_n(X)\to N_n(Y)}$ is a weak equivalence of simplicial sets. We denote by $\W_n^{\mathrm{Th}}$ the class of Thomason equivalences.

\end{definition}

\begin{paragr}\label{paragr:notationthom}

We usually make reference to Thomason equivalences in the notations of homotopic constructions induced by these equivalences. For example, we write $\Ho(n\Cat^{\Th})$ for the homotopy op-prederivator of $(n\Cat,\W_n^{\Th})$ and

We usually make reference to Thomason equivalences in the notations of homotopic constructions induced by these equivalences. For example, we write $\Ho(n\Cat^{\Th})$ for the homotopy op\nbd{}prederivator of $(n\Cat,\W_n^{\Th})$ and

\[

\gamma^{\Th} : n\Cat\to\Ho(n\Cat^{\Th})

\]

for the localization morphism. The reason is to avoid confusion with other weak equivalences on $n\Cat$which we will introduce later.

for the localization morphism. The reason is to avoid confusion with other weak equivalences on $n\Cat$that we will introduce later.

\end{paragr}

\begin{paragr}

By definition, the nerve functor induces a morphism of localizers ${N_n : (n\Cat,\W_n^{\Th})\to(\Psh{\Delta},\W_{\Delta})}$ and hence a morphism of op-prederivators

By definition, the nerve functor induces a morphism of localizers ${N_n : (n\Cat,\W_n^{\Th})\to(\Psh{\Delta},\W_{\Delta})}$ and hence a morphism of op\nbd{}prederivators

From Lemma \ref{lemma:dereq}, we have the following corollary.

From Lemma \ref{lemma:dereq}, we obtain the following corollary.

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

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

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.

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

...

...

@@ -219,13 +219,13 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\begin{remark}

Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$ by Ara and Maltsiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.

\end{remark}

By definition, for all $1\leq n \leq m \leq\omega$, the canonical inclusion ${n\Cat\hookrightarrow m\Cat}$ sends Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op-prederivator $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.

By definition, for all $1\leq n \leq m \leq\omega$, the canonical inclusion \[n\Cat\hookrightarrow m\Cat\] sends Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivator $\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})$.

\begin{proposition}

For all $1\leq n \leq m \leq\omega$, the canonical morphism

\[

\Ho(n\Cat^\Th)\to\Ho(m\Cat^{\Th})

\]

is an equivalence of op-prederivators.

is an equivalence of op\nbd{}prederivators.

\end{proposition}

\begin{proof}

This follows from Theorem \ref{thm:gagna} and the commutativity of the triangle

...

...

@@ -239,22 +239,23 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\section{Tensor product and oplax transformations}

Recall that $\oo\Cat$ can be equipped with a monoidal product $\otimes$, introduced by Al-Agl and Steiner in \cite{al1993nerves} and by Crans in \cite{crans1995combinatorial}, commonly referred to as the \emph{Gray tensor product}. The implicit reference for this section is \cite[Appendices A and B]{ara2016joint}.

\begin{paragr}

The Gray tensor product makes $\oo\Cat$ into a monoidal category for which the unit is the $\oo$-category $\sD_0$ (which is the terminal $\oo$-category). This monoidal category is \emph{not} symmetric but it is biclosed \cite[Theorem A.15]{ara2016joint}, meaning that there exist two functors

The Gray tensor product makes $\oo\Cat$ into a monoidal category for which the unit is the $\oo$\nbd{}category $\sD_0$ (which is the terminal $\oo$\nbd{}category). This monoidal category is \emph{not} symmetric but it is biclosed \cite[Theorem A.15]{ara2016joint}, meaning that there exist two functors

Hence, the $0$-cells of the $\oo$-category $\underline{\hom}_{\mathrm{oplax}}(Y,Z)$ are the $\oo$-functors $Y \to Z$.

Hence, the $0$-cells of the $\oo$\nbd{}category $\underline{\hom}_{\mathrm{oplax}}(Y,Z)$ are the $\oo$\nbd{}functors $Y \to Z$.

\end{paragr}

\begin{paragr}

Let $u,v : X \to Y$ be two $\oo$-functors. An \emph{oplax transformation} from $u$ to $v$ is a $1$-cell $\alpha$ of $\homoplax(X,Y)$ with source $u$ and target $v$. We usually use the double arrow notation \[

Let $u,v : X \to Y$ be two $\oo$\nbd{}functors. An \emph{oplax transformation} from $u$ to $v$ is a $1$-cell $\alpha$ of $\homoplax(X,Y)$ with source $u$ and target $v$. We usually use the double arrow notation \[

\alpha : u \Rightarrow v

\]

for oplax transformations. By adjunction, we have

...

...

@@ -264,7 +265,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\end{align*}

Hence, $\alpha : u \Rightarrow v$ can be encoded in the following two ways:

\begin{itemize}[label=-]

\item As an $\oo$-functor $\alpha : \sD_1\otimes X \to Y$ such that the following diagram

\item As an $\oo$\nbd{}functor $\alpha : \sD_1\otimes X \to Y$ such that the following diagram

\[

\begin{tikzcd}

X\ar[rd,"u"]\ar[d,"i_0^X"']&\\

...

...

@@ -272,8 +273,8 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

X \ar[ru,"v"']\ar[u,"i_1^X"]&

\end{tikzcd}

\]

where $i_0^X$ and $i_1^X$ are induced by the two $\oo$-functors $\sD_0\to\sD_1$ and where we implicitly used the isomorphism $\sD_0\otimes X \simeq X$, is commutative.

\item As an $\oo$-functor $\alpha : X \to\homlax(\sD_1,Y)$ such that the following diagram

where $i_0^X$ and $i_1^X$ are induced by the two $\oo$\nbd{}functors $\sD_0\to\sD_1$ and where we implicitly used the isomorphism $\sD_0\otimes X \simeq X$, is commutative.

\item As an $\oo$\nbd{}functor $\alpha : X \to\homlax(\sD_1,Y)$ such that the following diagram

\[

\begin{tikzcd}

& Y \\

...

...

@@ -281,15 +282,15 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

& Y

\end{tikzcd}

\]

where $\pi^Y_0$ and $\pi^Y_1$ are induced by the two $\oo$-functors $\sD_0\to\sD_1$ and where we implicitly used the isomorphism $\homlax(\sD_0,Y)\simeq Y$, is commutative.

where $\pi^Y_0$ and $\pi^Y_1$ are induced by the two $\oo$\nbd{}functors $\sD_0\to\sD_1$ and where we implicitly used the isomorphism $\homlax(\sD_0,Y)\simeq Y$, is commutative.

\end{itemize}

The $\oo$-category $\homlax(\sD_1,Y)$ is sometimes referred to as the $\oo$-category of cylinders in $Y$. An explicit description of this $\oo$-category can be found, for example, in \cite[Appendix A]{metayer2003resolutions}, \cite[Section 4]{lafont2009polygraphic} or \cite[Appendice B.1]{ara2016joint}.

The $\oo$\nbd{}category $\homlax(\sD_1,Y)$ is sometimes referred to as the $\oo$\nbd{}category of cylinders in $Y$. An explicit description of this $\oo$\nbd{}category can be found, for example, in \cite[Appendix A]{metayer2003resolutions}, \cite[Section 4]{lafont2009polygraphic} or \cite[Appendice B.1]{ara2016joint}.

\end{paragr}

\begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax transformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to those given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}.

Let $u, v : X \to Y$ two $\oo$-functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of:

Let $u, v : X \to Y$ two $\oo$\nbd{}functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of:

\begin{itemize}[label=-]

\item for every $0$-cell $x$ of $X$, a $1$-cell of $Y$

\[

...

...

@@ -297,7 +298,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\]

\item for every $n$-cell of $x$ of $X$ with $n>0$, an $(n+1)$-cell of $Y$

@@ -322,15 +323,15 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

In particular, when $n=1$ and $C$ and $D$ are thus (small) categories, an oplax transformation $u \Rightarrow v$ is nothing but a natural transformation from $u$ to $v$.

\end{example}

\begin{paragr}

Let $u : C \to D$ be an $\oo$-functor. There is an oplax transformation from $u$ to $u$, denoted by $1_u$, which is defined as

Let $u : C \to D$ be an $\oo$\nbd{}functor. There is an oplax transformation from $u$ to $u$, denoted by $1_u$, which is defined as

\[

(1_u)_{x}:=1_{u(x)}

\]

for every $n$-cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$-functor

for every $n$-cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$\nbd{}functor

\[

\sD_1\otimes C \overset{p\otimes i}{\longrightarrow}\sD_0\otimes D \simeq D,

\]

where $p$ is the only $\oo$-functor $\sD_1\to\sD_0$.

where $p$ is the only $\oo$\nbd{}functor $\sD_1\to\sD_0$.

\end{paragr}

\begin{paragr}

Let

...

...

@@ -348,7 +349,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\[

(\alpha\star f)_x :=\alpha_{f(x)}

\]

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. $\alpha\star f$).

for each cell $x$ of $B$) defines an oplax transformation from $g \circ u$ to $g \circ v$ (resp. $u \circ f$ to $v\circ f$) which we denote by $g\star\alpha$ (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

\[

...

...

@@ -360,28 +361,28 @@ 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 $\oo$\nbd{}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 $\oo$\nbd{}functors $u, v : C \to D$ are \emph{oplax homotopic} if they are equivalent for 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 $\mathrm{id}_C$.

\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.

Recall that we write$\gamma^{\Th} : \oo\Cat\to\ho(\oo\Cat^{\Th})$ the localization functor with respect to the Thomason equivalences.

\begin{lemma}\label{lemma:oplaxloc}

Let $u, v : C \to D$ be two $\oo$-functors. If there exists an oplax transformation $\alpha : u \Rightarrow v$, then $\gamma(u)=\gamma(v)$.

Let $u, v : C \to D$ be two $\oo$\nbd{}functors. If there exists an oplax transformation $\alpha : u \Rightarrow v$, then $\gamma^{\Th}(u)=\gamma^{\Th}(v)$.

\end{lemma}

\begin{proof}

This follows immediately from \cite[Théorème B.11]{ara2020theoreme}.

\end{proof}

From this lemma and the fact that Thomason equivalences are saturated (Corollary \ref{cor:thomsaturated}) we deduce the following proposition.

From this lemma and the fact that the Thomason equivalences are saturated (Corollary \ref{cor:thomsaturated}), we deduce the following proposition.

\begin{proposition}\label{prop:oplaxhmtpyisthom}

Every oplax homotopy equivalence is a Thomason equivalence.

\end{proposition}

\begin{paragr}\label{paragr:defrtract}

An $\oo$-functor $i : C \to D$ is an \emph{oplax deformation retract} if there exists an $\oo$-functor $r : C \to D$ such that:

An $\oo$\nbd{}functor $i : C \to D$ is an \emph{oplax deformation retract} if there exists an $\oo$\nbd{}functor $r : C \to D$ such that:

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

\item$r\circ i=\mathrm{id}_C$,

\item there exists an oplax transformation $\alpha : \mathrm{id}_B \Rightarrow i\circ r$.

...

...

@@ -440,7 +441,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\]

The existence of $\alpha' : \sD_1\otimes B' \to B'$ that makes the whole diagram commutes follows from the fact that the functor $\sD_1\otimes\shortminus$ preserves colimits. In particular, we have $\alpha' \circ(\sD_1\otimes i')= p \otimes i'$.

Now, notice that for any $\oo$-category $C$, the maps

Now, notice that for every $\oo$\nbd{}category $C$, the maps

\[

i^C_0 : C \to\sD_1\otimes C \text{ and } i^C_1 : C \to\sD_1\otimes C

\]

...

...

@@ -522,11 +523,11 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with