100 research outputs found
Strict Rezk completions of models of HoTT and homotopy canonicity
We give a new constructive proof of homotopy canonicity for homotopy type
theory (HoTT). Canonicity proofs typically involve gluing constructions over
the syntax of type theory. We instead use a gluing construction over a "strict
Rezk completion" of the syntax of HoTT. The strict Rezk completion is specified
and constructed in the topos of cartesian cubical sets. It completes a model of
HoTT to an equivalent model satisfying a saturation condition, providing an
equivalence between terms of identity types and cubical paths between terms.
This generalizes the ordinary Rezk completion of a 1-category
The univalence axiom for elegant Reedy presheaves
We show that Voevodsky's univalence axiom for intensional type theory is
valid in categories of simplicial presheaves on elegant Reedy categories. In
addition to diagrams on inverse categories, as considered in previous work of
the author, this includes bisimplicial sets and -spaces. This has
potential applications to the study of homotopical models for higher
categories.Comment: 25 pages; v2: final version, to appear in HH
CANONICITY AND HOMOTOPY CANONICITY FOR CUBICAL TYPE THEORY
Cubical type theory provides a constructive justification of homotopy type theory. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. We present in this article two canonicity results, both proved by a sconing argument: a homotopy canonicity result, every natural number is path equal to a numeral, even if we take away the equations defining the lifting operation on the type structure, and a canonicity result, which uses these equations in a crucial way. Both proofs are done internally in a presheaf model
Homotopy Canonicity for Cubical Type Theory
Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. In this paper we show by a sconing argument that if we remove these equations for the path lifting operation from the system, we still retain homotopy canonicity: every natural number is path equal to a numeral
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