nLab
cubical object

A cubical object in a category $C$ is a functor

$$F:{\square }^{\mathrm{op}}\to C$$F : \Box^{op} \to C

from the opposite category of the cube category $\square$ into $C$.

If $C=\mathrm{Set}$ then this is called a cubical set.

Functors

$$F:\square \to C$$F : \Box \to C

from the cube category itself are co-cubical objects in $C$.

See also simplicial object.