From Cubes to Twisted Cubes via Graph Morphisms in Type Theory
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Cube categories are used to encode higher-dimensional structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of such higher categories. Bezem, Coquand, and Huber (2014) have presented a constructive model of univalence using a specific cube category, which we call the "BCH category". Directed type theory examines the possibility that types could come with a notion of directed, not necessarily invertible, equality. Using presheaves on the BCH category with the usual Kan filling principles seems problematic if we want to model directed type theory since a notion of invertibility is built-in. To remedy this, we suggest a variation that we call "twisted cubes". Our strategy is to first develop several alternative (but equivalent) presentations of the BCH 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 simplices (tetrahedra).
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