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 nn-globe is the globular analog of the cellular nn-simplex. It is one of the basic geometric shapes for higher structures.

Definition

The cellular nn-globe G nG_n is the globular set represented by the object [n][n] in the globe category GG:

G n:=Hom G(,[n]). 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

nn-Category structure

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

GωCat G \to \omega Cat
[n]G n. [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.

Revised on May 17, 2012 10:53:26 by Urs Schreiber (89.204.153.6)