Clemens Berger, A Cellular Nerve for Higher Categories (Rev #3, 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 $(T(n))_{n\in \mathhb{N}_0}$ 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

$$\begin{array}{cc}{y}^{{\textstyle \prime}}& {y}^{{\textstyle \prime}{\textstyle \prime}}\\ \searrow & \swarrow \\ & y\end{array}$$ \array{~~ y^\prime&y^{\prime\prime}~~ y^\prime&&y^{\prime\prime} \\~~ \searrow&\swarrow~~ \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 *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.

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