for all subject to the relations (dropping obvious subscripts)
called the -source and -target maps (or similar), such that the globular identities hold: for all
The globular identities ensure that two sequences of boundary maps
with and for are equal if and only if their last term coincides; for all we have
For a globular set we may therefore write unambiguously
with the sequence of consecutive identity-assigning, source or target maps, respectively.
The presheaf definition can understood from the point of view of space and quantity: a globular set is a space characterized by the fact that and how it may be probed by mapping standard globes into it: the set assigned by a globular set to the standard -globe is the set of -globes in this space, hence the way of mapping a standard -globe into this spaces.
If to the globe category we add additional generating morphisms
satisfying the relations
we obtain the reflexive globe category, a presheaf on which is a reflexive globular set. In this case the morphism
is called the th identity assigning map; it satisfies the globular identities:
A presheaf on the full subcategory of the globe category containing only the integers through is called an -globular set or an -graph or an -graph. An -globular set may be identified with an -globular set which is empty above dimension .
Any strict 2-category or bicategory has an underlying 2-globular set. Likewise, any tricategory has an underlying 3-globular set. Globular sets can be used as underlying data for n-categories as well; see for instance Batanin ω-category.
The globular -globe is the globular set represented by , i.e. .
Also related is the notion of computad, which is similar to a globular set in some ways, but allows “formal composites” of -cells to appear in the sources and targets of -cells.
The definition is reviewed around def. 1.4.5, p. 49 of
R. Street, The petit topos of globular sets , JPAA 154 (2000) pp.299-315.
The definition of globular set, without using that term, is in 2.2 and 2.3 of the following paper:
The following paper constructs from the cubical case a strict globular -groupoid of a filtered space: