The globe category $G$ encodes one of the main geometric shapes for higher structures. Its objects are the standard cellular $n$-globes, and presheaves on it are globular sets.
It may also be called the globular category, although that term has other interpretations.
The globe category $G$ is the category whose objects are the non-negative integers and whose morphisms are generated from
for all $n \in \mathbb{N}$ subject to the relations (dropping obvious subscripts)
If we add the generating morphisms
subject to the relations
we obtain the reflexive globe category.
The globe category is used to define globular sets.
M. Roy has studied levels in the topos of reflexive globular sets in the context of F. W. Lawvere’s concept of Aufhebung (cf. Kennett-Riehl-Roy-Zaks(2011)).