The -dimensional cube, or simply -cube, is a generalisation of the ordinary cube (or -cube) to arbitrary dimensions. It comes in may guises.
As a cubical set
The standard cubical -cube is the cubical set represented (as a presheaf) by the object in the cube category.
As a topological space
The standard topological -cube is the space , where is the unit interval. The collection of topological cubes forms a topological cocubical set?.