## 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 $$\array{ y^\prime&y^{\prime\prime} \\ \searrow&\swarrow \\ &y }$$ =--