@@ -12,7 +12,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego

\[

\src(1_{x})=\trgt(1_{x})= x.

\]

The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs}$ f : G \to G'$ consists of maps $f_0 : G_0\to G'_0$ and $f_1 : G_1\to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they are the morphisms $f : G \to G'$ that are injective on objects and on arrows, i.e. such that $f_0 : G_0\to G_0'$ and $f_1 : G_1\to G'_1$ are injective.

The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs}$ f : G \to G'$ consists of maps $f_0 : G_0\to G'_0$ and $f_1 : G_1\to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they are the morphisms $f : G \to G'$ that are injective on objects and on arrows, i.e. such that $f_0 : G_0\to G_0'$ and $f_1 : G_1\to G'_1$ are injective.

There is a ``underlying reflexive graph'' functor

\[

...

...

@@ -31,7 +31,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego

of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$ and is denoted by $\ell(f)$. Composition is given by concatenation of chains.

\end{paragr}

\begin{lemma}

A category $C$ is free in the sense of \todo{ref} if and only if there exists a reflexive graph $G$ such that

A category $C$ is free in the sense of \ref{def:freeoocat} if and only if there exists a reflexive graph $G$ such that

\[

C \simeq L(G).

\]

...

...

@@ -42,7 +42,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego

Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the description of the arrows of $L(G)$ given in the previous paragraph shows that $C$ is free and that its set of generating $1$-cells is (isomorphic to) the non unital $1$-cells of $G$.

\end{proof}

\begin{remark}

Note that for a morphism of reflexive graphs $f : G \to G'$, the functor $L(f)$ is not necessarily rigid in the sense of \todo{ref} because generating $1$-cells may be sent to units.

In other words, a category is free on a graph if and only if it is free on a reflexive graph. The difference between these two notions is at the level of morphisms: there are more morphisms of reflexive graphs because (generating) $1$\nbd{}cells may be sent to units. Hence, for a morphism of reflexing graphs $f : G \to G'$, the induced functor $L(f)$ is not necessarily rigid in the sense of Definition \ref{def:rigidmorphism}.

\end{remark}

\begin{paragr}

There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq1}}$, the category of pre-sheaves on $\Delta_{\leq1}$. In particular, the canonical inclusion $i : \Delta_{\leq1}\rightarrow\Delta$ induces by pre-composition a functor

...

...

@@ -66,7 +66,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego

if $f_0 : X_0\to Y_0$ and $f_1 : X_1\to Y_1$ are monomorphisms and if all $n$-simplices of $X$ are degenerated for $n\geq2$, then $f$ is a monomorphism. A proof of this assertion is contained in \cite[Paragraph 3.4]{gabriel1967calculus}. The key argument is the Eilenberg-Zilber Lemma (Proposition 3.1 of op. cit.).

\end{proof}

\begin{paragr}

Let us denote by $N : \Psh{\Delta}\to\Cat$ (instead of $N_1$ as in Paragraph \todo{ref}) the usual nerve of categories and by $c : \Cat\to\Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an $n$-simplex of $N(C)$ is a chain

Let us denote by $N : \Psh{\Delta}\to\Cat$ (instead of $N_1$ as in Paragraph \ref{paragr:nerve}) the usual nerve of categories and by $c : \Cat\to\Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an $n$-simplex of $N(C)$ is a chain

@@ -200,7 +200,7 @@ From the previous proposition, we deduce the following very useful corollary.

i_!(C)\ar[r,"i_!(\gamma)"]& i_!(D).

\end{tikzcd}

\]

This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and \todo{ref}.

This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and Lemma \ref{lemma:hmtpycocartesianreedy}.

\end{proof}

\begin{paragr}

Actually, by working a little more, we obtain a more general result, which is stated in the propositon below. Let us say that a morphism of reflexive graphs, $\alpha : A \to B$, is \emph{quasi-injective on arrows} when for all $f$ and $g$ arrows of $A$, if

...

...

@@ -267,7 +267,7 @@ From the previous proposition, we deduce the following very useful corollary.

\ar[from=2-2,to=3-3,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny\textbf{3}}}", description]

\end{tikzcd}.

\]

What we want to prove is that the image by the functor $L$ of the pasting of squares \textcircled{\tiny\textbf{2}} and \textcircled{\tiny\textbf{3}} is homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor $L$ of square \textcircled{\tiny\textbf{3}} is homotopy cocartesian. Hence, all we have to show is that the image by $L$ of square \textcircled{\tiny\textbf{2}} is homotopy cocartesian. \todo{Parler du pasting lemma ?}On the other hand, we know that both morphisms

What we want to prove is that the image by the functor $L$ of the pasting of squares \textcircled{\tiny\textbf{2}} and \textcircled{\tiny\textbf{3}} is homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor $L$ of square \textcircled{\tiny\textbf{3}} is homotopy cocartesian. Hence, in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is that the image by $L$ of square \textcircled{\tiny\textbf{2}} is homotopy cocartesian. On the other hand, we know that both morphisms

\[

\coprod_{x \in E}F_x \to A \text{ and } A \to C

\]

...

...

@@ -308,10 +308,10 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

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 ?}

\end{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{Préciser que si A=B dans l'exemple précédent ça ne marche pas ?}

Note also that the hypothesis that $A\neq B$ was fundamental in the previous example as for $A=B$ the square is \emph{not} homotopy cocartesian.

@@ -78,7 +78,7 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para

For every $n \in\mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is oplax contractible.

\end{lemma}

\begin{proof}

Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}-cells for every $k$ such that $0\leq k<n$. These two $k$\nbd{}cells are parallel and are given by $\src_k(e_n)$ and $\trgt_k(e_n)$.

Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}cells for every $k$ such that $0\leq k<n$. These two $k$\nbd{}cells are parallel and are given by $\src_k(e_n)$ and $\trgt_k(e_n)$.

Let $r : \sD_0\to\sD_n$ be the $\oo$\nbd{}functor that points to $\trgt_0(e_n)$ (which means that $r=\langle\trgt_0(e_n)\rangle$ with the notations of \ref{paragr:defglobe}). Hence for every $k$\nbd{}cell $x$ of $\sD_n$, we have

\[

...

...

@@ -141,19 +141,25 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re

monomorphisms.

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

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

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

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

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

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

properties:

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

\item Every $k$\nbd{}cell of $\sS_n$ with $0\leq k <n$ is the image by

$j^+_n$ of a (unique) $k$\nbd{}cell of $\sD_n$\emph{and} the image by

$j^-_n$ of a (unique) $k$\nbd{}cell of $\sD_n$

\item Every non-trivial $n$\nbd{}cell of $\sS_n$ (there are only two of them) is the image either by

$j^+_n$ or by $j^-_n$ of a non-trivial $n$\nbd{}cell of $\sD_n$.

\end{enumerate}

Let $\varphi : \Or\to\sD_{n}$

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

%% properties:

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

%% \item Every $k$\nbd{}cell of $\sS_n$ with $0 \leq k <n$ is the image by

%% $j^+_n$ of a (unique) $k$\nbd{}cell of $\sD_n$ \emph{and} the image by

%% $j^-_n$ of a (unique) $k$\nbd{}cell of $\sD_n$

%% \item Every non-trivial $n$\nbd{}cell of $\sS_n$ (there are only two of them) is the image either by

%% $j^+_n$ or by $j^-_n$ of a non-trivial $n$\nbd{}cell of $\sD_n$.

%% \end{enumerate}

For convenience, let us write $h_n^+$ (resp.\ $h_n^-$) for the only generating $n$\nbd{}cell of $\sS_n$ contained in the image of $j^+_n$ (resp.\ $j_n^-$). The cells $h_n^+$ and $h_n^-$ are the only non-trivial $n$\nbd{}cells of $\sS_n$. We also write $\alpha_k$ for the principal cell of $\Or_k$ (see \ref{paragr:orientals}). This is the only non-trivial $k$\nbd{}cell of $\Or_k$.

Now, let $\varphi : \Or_k \to\sS_n$ be an $\oo$\nbd{}functor. There are several cases to distinguish.

\begin{description}

\item[Case $k<n$:] Since every generating cell of $\gamma$ of $\Or_k$ is of dimension lower or equal to $k$, the cell $\varphi(\gamma)$ is of dimension strictly lower than $n$. Since all cells of dimension strictly lower than $n$ are both in the image of $j^+_n$ and in the image of $j^-_n$, $\varphi$ obviously factors through $j^+_n$ (and $j^+_n$).

\item[Case $k=n$:] The image of $\alpha_n$ is either a non-trivial $n$\nbd{}cell of $\sS_n$ or a unit on a strictly lower dimensional cell. In the second situation, everything works like the case $k<n$. Now suppose for example that $\varphi(\alpha_n)$ is $h^+_n$, which is in the image of $j^+_n$. Since all of the other generating cells of $\Or_n$ are of dimension strictly lower than $n$, their images by $\varphi$ are also of dimension strictly lower than $n$ and hence, are all contained in the image of $j^+_n$. Altogether this proves that $\varphi$ factors through $j^+_n$. The case where $\varphi(\alpha_n)=h^-_n$ is symmetric.

\item[Case $k>n$:] Since $\sS_n$ is an $n$\nbd{}category, the image of $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If the dimension of $x$ is strictly lower than $n$, then everything works like in the case $k<n$. If not, this means that $x$ is a non-trivial $n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let $\gamma$ be a generator of $\Or_k$ of dimension $k-1$. Necessarily, we have $\varphi(\gamma)=1^n_y$ for $y$ an $n$\nbd{}cell of dimension non-greater than $n$ (with the convention that if the dimension of $y$ is $n$, then $1^ny:=y$). If the dimension

\end{description}

\end{proof}

From these two lemmas, follows the important proposition below.

@@ -383,7 +383,7 @@ We can now state the promised result, whose proof can be found in \cite[Section

We often say refer to the elements of the $n$-basis of a free $\oo$-category as the \emph{generating $n$-cells}. This sometimes leads to use the alternative terminology \emph{set of generating $n$-cells} instead of \emph{$n$-basis}.

\end{paragr}

\begin{definition}

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

Let $X$ and $Y$ be two free $\oo$-categories. An $\oo$-functor $f : X \to Y$ is \emph{rigid} if for every $n\geq0$ and every generating $n$-cell $x$ of $X$, $f(x)$ is a generating $n$-cell of $Y$.

\end{definition}

So far, we have not yet seen examples of free $\oo$-categories. In order to do that, we will explain in a further section a recursive way of constructing free $\oo$-categories; but first we take a little detour.