Varieties of Cubical Sets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10226)

Abstract

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.

References

  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)MathSciNetCrossRefMATHGoogle Scholar
  6. Grandis, M., Mauri, L.: Cubical sets and their site. Theory Appl. Categ. 11(8), 185–211 (2003). http://www.tac.mta.ca/tac/volumes/11/8/11-08abs.html MathSciNetMATHGoogle Scholar
  7. Grothendieck, A.: Pursuing stacks. Manuscript (1983). http://thescrivener.github.io/PursuingStacks/
  8. Maltsiniotis, G.: La théorie de l’homotopie de Grothendieck. Astérisque, vol. 301 (2005). https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/prstnew.pdf
  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 MathSciNetCrossRefMATHGoogle 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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