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.

Keywords

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.

Notes

Acknowledgements

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.

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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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