nLab
globe
Context
Higher category theory
higher category theory

Basic concepts
Basic theorems
Applications
Models
Morphisms
Functors
Universal constructions
Extra properties and structure
1-categorical presentations
Contents
Idea
The cellular $n$ -globe is the globular analog of the cellular $n$ -simplex . It is one of the basic geometric shapes for higher structures .

Definition
The cellular $n$ -globe $G_n$ is the globular set represented by the object $[n]$ in the globe category $G$ :

$G_n := Hom_G(-,[n])
\,.$

Examples
The 0-globe is the singleton set , the category with a single morphism.

The 1-globe is the interval category .

The 3-globe looks like this

Properties
$n$ -Category structure
There is a unique structure of a strict omega-category , an n-category in fact, on the $n$ -globe. This makes the collection of $n$ -globes arrange themselves into a co-globular $\omega$ -category , i.e. a functor

$G \to \omega Cat$

$[n] \mapsto G_n
\,.$

Relation to simplices
The orientals translate between simplices and globles.

References
See the references at strict omega-category and at oriental .