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Batanin’s -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 endowed with a map decreasing the degree by one and such that all fibers are linearly ordered.
Let be a tree.
A -sector of height is defined to be a cospan
denoted by where and are consecutive vertices in the linear order .
The source of a sector is defined to be where are consecutive vertices.
The target of a sector is defined to be where are consecutive vertices.