Cubical complex

In mathematics, a cubical complex (also called cubical set and Cartesian complex ) is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analogously to simplicial complexes and CW complexes in the computation of the homology of topological spaces.

Definitions
An elementary interval is a subset $$I\subsetneq\mathbf{R}$$ of the form


 * $$I = [l, l+1]\quad\text{or}\quad I=[l, l]$$

for some $$l\in\mathbf{Z}$$. An elementary cube $$Q$$ is the finite product of elementary intervals, i.e.


 * $$Q=I_1\times I_2\times \cdots\times I_d\subsetneq \mathbf{R}^d$$

where $$I_1,I_2,\ldots,I_d$$ are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube $$[0,1]^n$$ embedded in Euclidean space $$\mathbf{R}^d$$ (for some $$n,d\in\mathbf{N}\cup\{0\}$$ with $$n\leq d$$). A set $$X\subseteq\mathbf{R}^d$$ is a cubical complex (or cubical set) if it can be written as a union of elementary cubes (or possibly, is homeomorphic to such a set).

Related terminology
Elementary intervals of length 0 (containing a single point) are called degenerate, while those of length 1 are nondegenerate. The dimension of a cube is the number of nondegenerate intervals in $$Q$$, denoted $$\dim Q$$. The dimension of a cubical complex $$X$$ is the largest dimension of any cube in $$X$$.

If $$Q$$ and $$P$$ are elementary cubes and $$Q\subseteq P$$, then $$Q$$ is a face of $$P$$. If $$Q$$ is a face of $$P$$ and $$Q\neq P$$, then $$Q$$ is a proper face of $$P$$. If $$Q$$ is a face of $$P$$ and $$\dim Q=\dim P-1$$, then $$Q$$ is a facet or primary face of $$P$$.

Algebraic topology
In algebraic topology, cubical complexes are often useful for concrete calculations. In particular, there is a definition of homology for cubical complexes that coincides with the singular homology, but is computable.