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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 collection of trees with maps of graded sets commuting with defines a category , called the category of trees.
The following -construction is due to Batanin.
The finite ordinal we can regard as the 1-level tree with input edges. Hence the simplex category $\Delta\hookrightarrow
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.)
The following -construction is due to Batanin.
Let be a monomorphism. tree.
A is called to be -sector of height cartesian is defined to be a cospan if
is denoted a by pullback for all 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 level a trees. monomorphism.
(1) Any map is called to be is injective.cartesian if
(2) The inclusions correspond bijectively to cartesian subobjects of .
(3) is The a inclusions pullback for all . correspond bijectively to plain subtrees of with a specific choice of -sector for each input vertex of . (…)
(1) Let The category defined be by having as objects the level trees 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) (1) A Any mapglobular theory is defined injective. to be a category such that
(2) The inclusions correspond bijectively to cartesian subobjects of .
is (3) an The inclusion inclusions of a wide subcategory such that representable presheaves on restrict correspond bijectively to sheaves plain on subtrees of . with a specific choice of-sector for each input vertex of . (…)
(3) Presheaves on which restrict to sheaves on are called -models.
(1) The forgetful category functor 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 equivalence of categories.
is an inclusion of a wide subcategory such that representable presheaves on restrict to sheaves on .
(3) Presheaves on which restrict to sheaves on are called -models.
Let The forgetful functor and show that iff by writing as a colimit of representables.
is an equivalence of categories.
Let and show that iff by writing as a colimit of representables.