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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.
The following -construction is due to Batanin.
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 set of -sector is graded by the height of sectors.
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.
To have a source and a target for every sector of we adjoin in every but the highest degree a lest- and a greatest vertex serving as new minimum and maximum for the linear orders . We denote this new tree by and the set of its sectors by and obtain source- and target operators . This operators satisfy
as one sees in the following diagram depicting an “augmented” tree of height
which means that is an -graph (also called globular set).
Now let denote the globe category whose unique object in degree is , and let denotes the linear -level tree.
Then we have is the standard -globe. (Note that the previous diagram corresponds to the standard globe.)
Let be a monomorphism.
is called to be cartesian if
is a pullback for all .
Let be level trees.
(1) Any map is injective.
(2) The inclusions correspond bijectively to cartesian subobjects of .
(3) The inclusions correspond bijectively to plain subtrees of with a specific choice of -sector for each input vertex of . (…)
(1) The category defined by having as objects the level trees and as morphisms the maps between the associated -graphs. These morphisms are called immersions. This category shall be equipped with the structure of a site by defining the covering sieves by epimorphic families (of immersions). This site is called the globular site.
(2) A globular theory is defined to be a category such that
is an inclusion of a wide subcategory such that representable presheaves on restrict to sheaves on .
(3) Presheaves on which restrict to sheaves on , we call -models.
The forgetful functor
is an equivalence of categories