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 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.

Remarks

• The globe category is used to define globular sets.

• M. Roy has studied levels in the topos of reflexive globular sets? in the context of F. W. Lawvere‘s concept of Aufhebung (cf. Kennett-Riehl-Roy-Zaks(2011)).

Related concepts

• globular set

• simplex category

• cube category

• cell 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