globe category

The globe category


The globe category GG encodes one of the main geometric shapes for higher structures. Its objects are the standard cellular nn-globes, and presheaves on it are globular sets.

It may also be called the globular category, although that term has other interpretations.


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

σ n:[n][n+1] \sigma_n : [n] \to [n+1]
τ n:[n][n+1] \tau_n : [n] \to [n+1]

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

σσ=τσ \sigma\circ \sigma = \tau \circ \sigma
στ=ττ \sigma\circ \tau = \tau \circ \tau

The reflexive globe category

If we add the generating morphisms

ι n:[n+1][n] \iota_n : [n+1] \to [n]

subject to the relations

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

we obtain the reflexive globe category.



  • 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 (