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UNIVERSES AND UNIVALENCE IN HOMOTOPY TYPE THEORY

Published online by Cambridge University Press:  15 July 2019

JAMES LADYMAN  and
STUART PRESNELL
JAMES LADYMAN*
Affiliation:
University of Bristol
STUART PRESNELL*
Affiliation:
University of Bristol
*
*DEPARTMENT OF PHILOSOPHY COTHAM HOUSE UNIVERSITY OF BRISTOL BRISTOL BS6 6JL, UK E-mail: james.ladyman@bristol.ac.uk
†DEPARTMENT OF PHILOSOPHY COTHAM HOUSE UNIVERSITY OF BRISTOL BRISTOL BS6 6JL, UK E-mail: stuart.presnell@bristol.ac.uk
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Abstract

The Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory (HoTT) research programme. It is said by Steve Awodey that Univalence embodies mathematical structuralism, and that Univalence may be regarded as ‘expanding the notion of identity to that of equivalence’. This article explores the conceptual, foundational and philosophical status of Univalence in Homotopy Type Theory. It extends our Types-as-Concepts interpretation of HoTT to Universes, and offers an account of the Univalence axiom in such terms. We consider Awodey’s informal argument that Univalence is motivated by the principle that reasoning should be invariant under isomorphism, and we examine whether an autonomous and rigorous justification along these lines can be given. We consider two problems facing such a justification. First, there is a difference between equivalence and isomorphism and Univalence must be formulated in terms of the former. Second, the argument as presented cannot establish Univalence itself but only a weaker version of it, and must be supplemented by an additional principle. The article argues that the prospects for an autonomous justification of Univalence are promising.

Keywords

03Homotopy Type TheoryequivalenceisomorphismUnivalenceAwodeystructuralismuniverses

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 3 , September 2019 , pp. 426 - 455
Copyright
Copyright © Association for Symbolic Logic 2019 

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