symmetric monoidal (∞,1)-category of spectra
n-category = (n,n)-category
n-groupoid = (n,0)-category
The notion of globular operads is a variant of that of operads on which certain algebraic notions of higher category are based. The notion was introduced by Batanin; a globular operad is also called a Batanin operad.
A globular operad gives rise to a monad on the category of globular sets; one example is the free strict ω-category monad on globular sets. The monads which so arise may be characterized precisely as cartesian monads on globular sets over (itself a cartesian monad). This means that they are also examples of generalized multicategories relative to .
A globular collection is a globular set equipped with a map
to , the underlying globular set of the free strict -category on the terminal globular set. Hence the category is the slice category
and the requisite map is given by the composite
where denotes multiplication of the monad . The monoidal unit is the collection where is the unit of , and the associativity and unit constraints may be defined by means of universal properties, taking advantage of the fact that is cartesian.
and the multiplication and unit for may be worked out from the multiplication and unit for the globular operad .
A more concrete description of may be worked out in terms of a concrete description of the free strict -category . To describe this, first notice that every element of , which is essentially a pasting diagram built up out of globes of , can be drawn as a globular set which we denote as . The globes of are instances of globular cells as they appear in the pasting diagram , and their sources and targets are then also instances of cells in . (Batanin describes in terms of trees, and the globular set is given formally in the tree language.)
Similarly, we can think of an element of as a pasting diagram built out of globes in , and such a pasting diagram can be thought of as having an underlying shape given by an pasting diagram in , together with a labeling of the pasting cells in by elements on . The labeling is in fact just a morphism of globular sets. Therefore we have an explicit formula for the set of -cells of :
and similarly, for a globular operad with underlying collection ,
The category of operators of a globular operad is (the syntactic category of) a homogeneous globular theory and every globular operad is characterized by its globular theory. See there for more details
The category of strict ω-categories is equivalent to that of algebras over the terminal globular operad. Hence it is the full subcategory of that of ω-graphs which satisfy the Segal condition with respect to the canonical inclusion that defines its globular theory: we have a pullback
As a refinement of the above example:
In the Batanin (or Leinster) theory of -categories, there is a universal contractible globular operad , where each element is thought of as a way of (weakly) pasting together the underlying shape . The contractibility implies that for every two different ways of pasting together the same shape, i.e., two elements such that and such that and have the same source and have the same target, there is an -cell in mediating between them, with source and target , and which maps to the identity -cell on .
A Batanin ω-category is a globular set with a -algebra structure.
A review and characterization in terms of globular theories is in section 1 of