A parametricity-based formalization of semi-simplicial and semi-cubical sets
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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2023
- DOI:
- 10.48550/arXiv.2401.00512
- arXiv:
- arXiv:2401.00512
- Bibcode:
- 2024arXiv240100512H
- Keywords:
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- Computer Science - Logic in Computer Science;
- F.4.1
- E-Print:
- Associated formalization in Coq at https://github.com/artagnon/bonak