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DOI: 10.4230/LIPIcs.TYPES.2019.5
From Cubes to Twisted Cubes via Graph Morphisms in Type Theory

Authors: Gun Pinyo and Nicolai Kraus

Published in: LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Abstract
Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of higher groupoids. Bezem, Coquand, and Huber [Bezem et al., 2014] have presented a constructive model of univalence using a specific cube category, which we call the BCH cube category. The higher categories encoded with the BCH cube category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible. This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH cube category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).
Cite as

Gun Pinyo and Nicolai Kraus. From Cubes to Twisted Cubes via Graph Morphisms in Type Theory. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{pinyo_et_al:LIPIcs.TYPES.2019.5,
  author =	{Pinyo, Gun and Kraus, Nicolai},
  title =	{{From Cubes to Twisted Cubes via Graph Morphisms in Type Theory}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-158-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{175},
  editor =	{Bezem, Marc and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.5},
  URN =		{urn:nbn:de:0030-drops-130694},
  doi =		{10.4230/LIPIcs.TYPES.2019.5},
  annote =	{Keywords: homotopy type theory, cubical sets, directed equality, graph morphisms}
}
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