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Univalence for inverse diagrams and homotopy canonicity

Published online by Cambridge University Press:  24 November 2014

MICHAEL SHULMAN
MICHAEL SHULMAN*
Affiliation:
University of San Diego, San Diego, CA, U.S.A. E-mail: shulman@sandiego.edu
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Abstract

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We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky's univalent model in simplicial sets, we obtain new models of univalence in a number of (∞, 1)-toposes; this answers a question raised at the Oberwolfach workshop on homotopical type theory. Second, by gluing the syntactic category of univalent type theory along its global sections functor to groupoids, we obtain a partial answer to Voevodsky's homotopy-canonicity conjecture: in 1-truncated type theory with one univalent universe of sets, any closed term of natural number type is homotopic to a numeral.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 5: From type theory and homotopy theory to Univalent Foundations of Mathematics , June 2015 , pp. 1203 - 1277
Copyright
Copyright © Cambridge University Press 2014 

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