## 1. Globular theories and cellular nerves Contents: Batanin's $\omega$-operads are described by their operator categories which are called *globular theories*. +-- {: .num_definition} ###### Definition (finite planar level tree) 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. +-- {: .num_lemma} ###### 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 $$\array{ y^\prime&&y^{\prime\prime} \\ \searrow&&\swarrow \\ &y }$$ 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 [[nLab:globular set]]). Now let $G$ denote the [[nLab: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. =--