# Cubical complex

In mathematics, a **cubical complex** (also called **cubical set** and **Cartesian complex**^{[1]}) 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

[edit]An **elementary interval** is a subset of the form

for some . An **elementary cube** is the finite product of elementary intervals, i.e.

where are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube embedded in Euclidean space (for some with ).^{[2]} A set 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).^{[3]}

### Related terminology

[edit]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 , denoted . The dimension of a cubical complex is the largest dimension of any cube in .

If and are elementary cubes and , then is a **face** of . If is a face of and , then is a **proper face** of . If is a face of and , then is a **facet** or **primary face** of .

## Algebraic topology

[edit]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.

## See also

[edit]## References

[edit]**^**Kovalevsky, Vladimir. "Introduction to Digital Topology Lecture Notes". Archived from the original on 2020-02-23. Retrieved November 30, 2021.**^**Werman, Michael; Wright, Matthew L. (2016-07-01). "Intrinsic Volumes of Random Cubical Complexes".*Discrete & Computational Geometry*.**56**(1): 93–113. arXiv:1402.5367. doi:10.1007/s00454-016-9789-z. ISSN 0179-5376.**^**Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004).*Computational Homology*. New York: Springer. ISBN 9780387215976. OCLC 55897585.

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