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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 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).
This -construction is due to Batanin.
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 , for let denotes the linear -level tree.
Then we have is the standard -globe.