Spahn Clemens Berger, A Cellular Nerve for Higher Categories (Rev #7, changes)

Lemma and Definition ($\omega$-graph of sectors of a tree)

Let $T$ be a tree.

A $T$-sector of height $k$ is defined to be a cospan

denoted by $(y;y^\prime,y^{\prime\prime})$ where $y\in T(k)$ and $y\lt y^{\prime\prime}$ are consecutive vertices in the linear order $i_T^{-1}(y)$.

The set $GT$ of $T$-sector is graded by the height of sectors.

The source of a sector $(y;y^\prime,y^{\prime\prime})$ is defined to be $(i(y);x,y)$ where $x,y$ are consecutive vertices.

The target of a sector $(y;y^\prime,y^{\prime\prime})$ is defined to be $(i(y);y,z)$ where $y,z$ are consecutive vertices.

To have a source and a target for every sector of $T$ we adjoin in every but the highest degree a lest- and a greatest vertex serving as new minimum and maximum for the linear orders $i^{-1}(x)$. We denote this new tree by $\overline T$ and the set of its sectors by $T_*:=G\overline T(k)$ and obtain source- and target operators $s,t:T_*\to T_*$. This operators satisfy

as one sees in the following diagram depicting an “augmented” tree of height $3$

which means that $T_*$ is an $\omega$-graph (also called globular set).

Now let $G$ denote the globe category whose unique object in degree $n\in \mathbb{N}$ n n\in \mathbb{N} is $n_G$ , for and let$nn\in \mathbb{N}$ n\in \mathbf{n} \mathbb{N} let denotes the linear$nn$ \mathbf{n} n denotes the linear $n$-level tree.

Then we have $\mathbf{n}_*\simeq G(-,n_G)$ is the standard $n$-globe. (Note that the previous diagram corresponds to the standard $3$ globe.)

Lemma

Let $S,T$ be level trees.

(1) Any map $S_*\to T_*$ is injective.

(2) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to cartesian subobjects of $T_*$.

(3) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to plain subtrees of $T$ with a specific choice of $T$-sector for each input vertex of $S$. (…)