# 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

${\sigma }_{n}:\left[n\right]\to \left[n+1\right]$\sigma_n : [n] \to [n+1]
${\tau }_{n}:\left[n\right]\to \left[n+1\right]$\tau_n : [n] \to [n+1]

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

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

### The reflexive globe category

If we add the generating morphisms

${\iota }_{n}:\left[n+1\right]\to \left[n\right]$\iota_n : [n+1] \to [n]

subject to the relations

$\iota \circ \sigma =\mathrm{Id}$\iota \circ \sigma = \mathrm{Id}
$\iota \circ \tau =\mathrm{Id}\phantom{\rule{thinmathspace}{0ex}}.$\iota \circ \tau = \mathrm{Id} \,.

we obtain the reflexive globe category.

## Remarks

category: category

Revised on November 1, 2012 03:16:40 by Urs Schreiber (82.169.65.155)