The $n$-dimensional cube, or simply $n$-cube, is a generalisation of the ordinary cube (or $3$-cube) to arbitrary dimensions. It comes in many guises.

The standard cubical $n$-cube is the cubical set represented (as a presheaf) by the object $[n]$ in the cube category.

The standard topological $n$-cube is the space $[0,1]^n$, where $[0,1]$ is the unit interval. The collection of topological cubes forms a topological cocubical set?.

cubical set

cubical type theory