Clemens Berger, A Cellular Nerve for Higher Categories (Rev #5)

Contents:

Batanin’s $\omega$-operads are described by their operator categories which are called *globular theories*.

A *finite planar level tree* ( or for short just a *tree*) is a graded set $(T(n))_{n\in \mathbb{N}_0}$ endowed with a map $i_T: T_{\gt 0}$ decreasing the degree by one and such that all fibers $i_T^{-1}(x)$ are linearly ordered.

The following ${}_*$-construction is due to Batanin.

Let $T$ be a tree.

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

$\array{
y^\prime&&y^{\prime\prime}
\\
\searrow&&\swarrow
\\
&y
}$

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.

$\array{
&y^\prime&&y^{\prime\prime}
\\
& \searrow&&\swarrow
\\
x&&y&&z
\\
\searrow&&\downarrow^i&&\swarrow
\\
&&i(y)
}$

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

$s\circ s=s\circ t$

$t\circ t =t\circ s$

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

$\array{
T(3)&&&y^\prime&&y^{\prime\prime}
\\
&&& \searrow&&\swarrow
\\
T(2)&&x&&y&&z
\\
&&\searrow&&\downarrow^i&&\swarrow
\\
T(1)&&u&&v&&w
\\
&&\searrow&&\downarrow^i&&\swarrow
\\
T(0)&&&&r
}$

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

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

Then we have $\mathbf{n}_*\simeq G(-,n_G)$ is the standard $n$-globe.