Clemens Berger, A Cellular Nerve for Higher Categories (Rev #4, changes)

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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 $(T(n){)}_{n\in mathhb{\mathbb{N}}_{0}{N}_{0}}$ (T(n))_{n\in~~ \mathhb{N}_0}~~ \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.

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 $i_T^{-1}$.

The set$GT$ *source of a sector $(y;y^\prime,y^{\prime\prime})$*~~ is~~ of~~ defined~~~~ to~~~~ be~~$(Ti(y);x,y)$~~ (i(y);x,y)~~ T~~ ~~ -sector~~ where~~ is graded by the height of sectors.~~$x,y$~~~~ are consecutive vertices.~~

The * target source of a sector$(y;y^\prime,y^{\prime\prime})$* is defined to be $(i(y);\mathrm{yx},\mathrm{zy})$

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

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* $s,t:T_*\to G\overline T_*$. 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).

This $*$-construction is due to Batanin.