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

Published online by Cambridge University Press:  23 June 2025

Hugo Herbelin Open the ORCID record for Hugo Herbelin [Opens in a new window]  and
Ramkumar Ramachandra Open the ORCID record for Ramkumar Ramachandra [Opens in a new window]
Hugo Herbelin*
Affiliation:
Université Paris Cité, Inria, CNRS, IRIF, Paris, France
Ramkumar Ramachandra
Affiliation:
Université Paris Cité (2020-2022), Unaffiliated, Paris, France
*
Corresponding author: Hugo Herbelin; Email: Hugo.Herbelin@inria.fr
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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 their 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.

Keywords

semi-simplicial setssemi-cubical setsparametricityCoqformalization

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Type
Special Issue: Advances in Homotopy type theory
Information
Mathematical Structures in Computer Science , Volume 35 , 2025 , e13
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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