A parametricitybased formalization of semisimplicial and semicubical sets
Abstract
Semisimplicial and semicubical sets are commonly defined as presheaves over respectively, the semisimplex or semicube category. Homotopy Type Theory then popularized an alternative definition, where the set of nsimplices or ncubes are instead regrouped into the families of the fibers over their faces, leading to a characterization we call indexed. Moreover, it is known that semisimplicial and semicubical 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 semisimplicial and semicubical sets, and fully formalize it in Coq.
 Publication:

arXiv eprints
 Pub Date:
 December 2023
 DOI:
 10.48550/arXiv.2401.00512
 arXiv:
 arXiv:2401.00512
 Bibcode:
 2024arXiv240100512H
 Keywords:

 Computer Science  Logic in Computer Science;
 F.4.1
 EPrint:
 Associated formalization in Coq at https://github.com/artagnon/bonak