We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, meaning that products of cubical sets correspond to products of homotopy types.
CITATION STYLE
Buchholtz, U., & Morehouse, E. (2017). Varieties of cubical sets. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 10226 LNCS, pp. 77–92). Springer Verlag. https://doi.org/10.1007/978-3-319-57418-9_5
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