## 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 $(T(n))_{n\in \mathhb{N}_0}$ 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. =-- +-- {: .num_remark} ###### 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 $i_T^{-1}$. 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. $$\array{ &y^\prime&&y^{\prime\prime} \\ & \searrow&&\swarrow \\ x&&y&&z \\ \searrow&&\downarrow^i&&\swarrow \\ &&i(y) }$$ =--