Cubical stuff

Although simplicial stuff is the standard for algebraic topology, “cubical stuff”—the cube category , cubical sets , cubical objects in a category, and so on—offer an alternative. Cubes are also one of the main geometric shapes for higher category theory.

Literature

General

  • Grandis & Mauri, 2003: Cubical sets and their site (pdf)
  • Buchholtz & Morehouse, 2017: Varieties of cubical sets (doi, arxiv)
  • Newstead, unpublished notes: Cubical sets (pdf)
    • Sec. 1: Helpful summary of equivalent definitions of cube category
    • Sec. 2: Cubical sets and their geometric realization

Homotopy

  • Brown & Higgins, 1981: On the algebra of cubes (doi, pdf)
    • Brown, Higgins, Sivera, 2011: Nonabelian algebraic topology (doi, pdf, nLab ), Ch. 11: The cubical classifying space of a crossed complex
    • Commentary on MO, including by Ronnie Brown
  • Jardine, 2002, unpublished: Cubical homotopy theory: a beginning (pdf)
    • Jardine, 2006: Categorical homotopy theory (pdf), especially Sec. 3: Cubical sets: basic properties

Homology

  • Kaczynski et al, 2004: Computational homology (doi)
    • Ch. 2: Cubical homology
    • Sec. 11.2: Comparison of cubical and simplicial complexes