A \emph{morphism of $\oo$-graphs}$f : X \to Y$ is a sequence $(f_n : X_n \to Y_n)_{n \in\mathbb{N}}$ of maps that is compatible with source and targets, i.e.\ for every $n$\nbd{}cell $x$ of $X$ with $n >0$, we have

A \emph{morphism of $\oo$-graphs}$f : X \to Y$ is a sequence $(f_n : X_n \to Y_n)_{n \in\mathbb{N}}$ of maps that is compatible with source and target, i.e.\ for every $n$\nbd{}cell $x$ of $X$ with $n >0$, we have

\[

f_{n-1}(\src(x))=\src(f_n(x))\text{ and } f_{n-1}(\trgt(x))=\trgt(f_n(x)).

\]

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@@ -114,7 +114,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n

\]

whenever these equations make sense.

A \emph{morphism of $\oo$-magmas}$f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and composition, i.e.\ for every $n$\nbd{}cell $x$, we have

A \emph{morphism of $\oo$-magmas}$f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have

\[

f(1_x)=1_{f(x)},

\]

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...

@@ -191,7 +191,10 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

The functors $\tau^{s}_{\leq n}$ and $\tau^{i}_{\leq n}$ are respectively referred to as the \emph{stupid truncation functor} and the \emph{intelligent truncation functor}.

The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$\nbd{}categories whose $k$\nbd{}cells for $k >n$ are all units.

The functor $\iota_n$ is fully faithful and preserves both limits and

colimits; in regards to these properties, we often identify $n\Cat$ with the

essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$

spanned by the $\oo$\nbd{}categories whose $k$\nbd{}cells for $k >n$ are all units.

\end{paragr}

\begin{paragr}

For $n \geq0$, we define the $n$\nbd{}skeleton functor $\sk_n : \oo\Cat\to\oo\Cat$ as

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@@ -200,16 +203,22 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

\]

This functor preserves both limits and colimits. For an $\oo$\nbd{}category

$C$, $\sk_n(C)$ is the sub-$\oo$\nbd{}category of $C$ generated by the

$k$\nbd{}cells of $C$ with $k\leq n$ in an obvious sense. In particular, we have a canonical filtration

$k$\nbd{}cells of $C$ with $k\leq n$ in an obvious sense. It is also

convenient to define $\sk_{-1}(C)$ to be the empty $\oo$\nbd{}category

is $C$ and the universal arrow $\sk_{n}(C)\to C$ is given by the co-unit of the adjunction $\tau^{s}_{\leq n}\dashv\iota_n$.

\end{lemma}

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@@ -217,7 +226,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo

For $n \in\mathbb{N}$, the \emph{$n$\nbd{}globe}$\sD_n$ is the $n$\nbd{}category that has:

\begin{itemize}[label=-]

\item exactly one non-trivial $n$\nbd{}cell, which we refer to as the \emph{principal $n$\nbd{}cell} of $\sD_n$, and which we denote by $e_n$,

\item exactly two non-trivial $k$\nbd{}cells for $k<n$; these $k$\nbd{}cells being parallel and given by the $k$\nbd{}source and the $k$\nbd{}target of $e_n$.

\item exactly two non-trivial $k$\nbd{}cells for every $k<n$; these $k$\nbd{}cells being parallel and given by the $k$\nbd{}source and the $k$\nbd{}target of $e_n$.

\end{itemize}

This completely describes the $n$\nbd{}category $\sD_n$ as no non-trivial composition can occur. Here are pictures in low dimension: