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arXiv · 2401.00512

A parametricity-based formalization of semi-simplicial and semi-cubical sets

Hugo Herbelin·Ramkumar Ramachandra
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Abstract

Semi-simplicial and semi-cubical sets are commonly defined as presheaves over respectively, the semi-simplex or semi-cube category. Homotopy Type Theory then popularized an alternative definition, where the set of n-simplices or n-cubes are instead regrouped into the families of the fibers over their faces, leading to a characterization we call indexed. Moreover, it is known that semi-simplicial and semi-cubical sets are related to iterated Reynolds parametricity, respectively in its unary and binary variants. We exploit this correspondence to develop an original uniform indexed definition of both augmented semi-simplicial and semi-cubical sets, and fully formalize it in Coq.

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Hugo Herbelin, Ramkumar Ramachandra. 2023-12-31. A parametricity-based formalization of semi-simplicial and semi-cubical sets. https://doi.org/10.1017/s096012952500009x

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