eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-10-12
202
220
10.4230/LIPIcs.TYPES.2014.202
article
Nominal Presentation of Cubical Sets Models of Type Theory
Pitts, Andrew M.
The cubical sets model of Homotopy Type Theory introduced by Bezem, Coquand and Huber uses a particular category of presheaves. We show that this presheaf category is equivalent to a category of sets equipped with an action of a monoid of name substitutions for which a finite support property holds. That category is in turn isomorphic to a category of nominal sets equipped with operations for substituting constants 0 and 1 for names. This formulation of cubical sets brings out the potentially useful connection that exists between the homotopical notion of path and the nominal sets notion of name abstraction. The formulation in terms of actions of monoids of name substitutions also encompasses a variant category of cubical sets with diagonals, equivalent to presheaves on Grothendieck's "smallest test category." We show that this category has the pleasant property that path objects given by name abstraction are exponentials with respect to an interval object.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol039-types2014/LIPIcs.TYPES.2014.202/LIPIcs.TYPES.2014.202.pdf
models of dependent type theory
homotopy type theory
cubical sets
nominal sets
monoids