Varieties of Cubical Sets

  • Ulrik BuchholtzEmail author
  • Edward Morehouse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10226)


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.


Function Symbol Monoidal Category Homotopy Theory Syntactic Category Structural Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We wish to thank the members of the HoTT group at Carnegie Mellon University for many fruitful discussions, in particular Steve Awodey who has encouraged the study of cartesian cube categories since 2013 and who has been supportive of our work, as well as Bob Harper who has also been very supportive. Additionally, we deeply appreciate the influence of Bas Spitters who inspired us with a seminar presentation of a different approach to showing that \(\mathbb {C}_{(\mathrm {wec},{\cdot })}\) is a strict test category.

The authors gratefully acknowledge the support of the Air Force Office of Scientific Research through MURI grant FA9550-15-1-0053. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the AFOSR.


  1. Angiuli, C., Harper, R., Wilson, T.: Computational higher-dimensional type theory. In: POPL 2017: Proceedings of the 44th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. ACM (2017, to appear)Google Scholar
  2. Awodey, S.: A cubical model of homotopy type theory (2016). arXiv:1607.06413. Lecture notes from a series of lectures for the Stockholm Logic group
  3. Bezem, M., Coquand, T., Huber, S.: A model of type theory in cubical sets. In: 19th International Conference on Types for Proofs and Programs, LIPIcs, Leibniz Leibniz International Proceedings in Informatics, vol. 26, pp. 107–128 (2014). doi: 10.4230/LIPIcs.TYPES.2013.107. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern
  4. Cohen, C., Coquand, T., Huber, S., Mörtberg, A.: Cubical type theory: a constructive interpretation of the univalence axiom. In: 21st International Conference on Types for Proofs and Programs, LIPIcs, Leibniz International Proceedings in Informatics (2016, to appear). Schloss Dagstuhl. Leibniz-Zent. Inform., WadernGoogle Scholar
  5. Gehrke, M., Walker, C.L., Walker, E.A.: Normal forms and truth tables for fuzzy logics. Fuzzy Sets Syst. 138(1), 25–51 (2003). doi: 10.1016/S0165-0114(02)00566-3. Selected papers from the 21st Linz Seminar on Fuzzy Set Theory (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Grandis, M., Mauri, L.: Cubical sets and their site. Theory Appl. Categ. 11(8), 185–211 (2003). MathSciNetzbMATHGoogle Scholar
  7. Grothendieck, A.: Pursuing stacks. Manuscript (1983).
  8. Maltsiniotis, G.: La théorie de l’homotopie de Grothendieck. Astérisque, vol. 301 (2005).
  9. Maltsiniotis, G.: La catégorie cubique avec connexions est une catégorie test stricte. Homology Homotopy Appl. 11(2), 309–326 (2009). doi: 10.4310/HHA.2009.v11.n2.a15 MathSciNetCrossRefzbMATHGoogle Scholar
  10. Mauri, L.: Algebraic theories in monoidal categories. Unpublished preprint (2005)Google Scholar
  11. Quillen, D.: Higher algebraic K-theory: I. In: Bass, H. (ed.) Higher K-Theories. LNM, vol. 341, pp. 85–147. Springer, Berlin (1973). doi: 10.1007/BFb0067053 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of PhilosophyCarnegie Mellon UniversityPittsburghUSA
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Carnegie Mellon School of Computer SciencePittsburghUSA
  4. 4.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA

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