# The globe category

## Idea

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.

## Definition

The globe category $G$ is the category whose objects are the non-negative integers and whose morphisms are generated from

$\sigma_n : [n] \to [n+1]$
$\tau_n : [n] \to [n+1]$

for all $n \in \mathbb{N}$ subject to the relations (dropping obvious subscripts)

$\sigma\circ \sigma = \tau \circ \sigma$
$\sigma\circ \tau = \tau \circ \tau$

### The reflexive globe category

If we add the generating morphisms

$\iota_n : [n+1] \to [n]$

subject to the relations

$\iota \circ \sigma = \mathrm{Id}$
$\iota \circ \tau = \mathrm{Id} \,.$

we obtain the reflexive globe category.

## Reference

• C. Kennett, E. Riehl, M. Roy, M. Zaks, Levels in the toposes of simplicial sets and cubical sets , JPAA 215 no.5 (2011) pp.949-961. (preprint)

category: category

Revised on December 27, 2014 15:14:55 by Thomas Holder (127.0.0.1)