Batanin’s $\omega$-operads are described by their operator categories which are called globular theories.

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.

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