nLab
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 integers and whose morphisms are generated from

$${\sigma }_n:[n]\to [n+1]$$\sigma_n : [n] \to [n+1]

$${\tau }_n:[n]\to [n+1]$$\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 \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

$${\iota }_n:[n+1]\to [n]$$\iota_n : [n+1] \to [n]

subject to the relations

$$\iota \circ \sigma =\mathrm{Id}$$\iota \circ \sigma = \mathrm{Id}

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

we obtain the reflexive globe category.

Remarks

• The globe category is used to define globular sets.

category: category