nLab cubical infinity-groupoid

Idea

The cubical analogue of a simplicial infinity-groupoid.

Definition

Let $\Box$ denote a category of cubes.

Definition

A cubical $\infty$-groupoid or cubical anima is an (∞,1)-functor

$$X \colon \Box^{op} \to \infty\mathrm{Grpd}$$

from the opposite category of $\Box$ into the (∞,1)-category ∞Grpd of ∞-groupoids.

Definition

The $(\infty,1)$-category of cubical $\infty$-groupoids and morphisms between them is the (∞,1)-category of (∞,1)-functors

$$\infty\mathrm{Grpd}^{\Box^{op}} = Func_\infty(\Box^{op}, \infty\mathrm{Grpd}) \,.$$

Cubical $\infty$-groupoids can be modeled by cubical-simplicial sets.

Related concepts

• infinity-groupoid

• simplicial infinity-groupoid

• (infinity,1)-functor

• category of cubes

• cubical set

• cubical object

• cubical object in an (infinity,1)-category

• triangulated type theory

• cubical-simplicial set