Universes and Univalence in Homotopy Type Theory

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.

What Types Should Not Be†

“…As was discussed in the previous section, it appears that the typesas-concepts interpretation can only serve as an incompletely specified informal semantics for homotopy type theory, at least in its current stage of development. Ladyman and Presnell (2017) do not provide a clear interpretation of universes and type formers, and, besides never addressing higher inductive types, and they do not show that the types-as-concepts interpretation can validate univalence.…”

“…On the basis of their representation of universes as domains of discourse, Ladyman and Presnell (2017) support a description of univalence as the claim that all domains of discourse are univalent and that of equivalent types as a single mathematical concept under multiple different presentations ( §5.1). However, as Ladyman and Presnell (2017, §5.1) admit, to justify such a view there is more to be done.…”

“…In their third paper, Ladyman and Presnell (2017) extend their types-as-concepts interpretation to universes and offer an account of the univalence axiom in such terms. To that end they must explain how we may understand universes as concepts and how we may justify univalence as a claim about concepts.…”

“…To that end they must explain how we may understand universes as concepts and how we may justify univalence as a claim about concepts. Ladyman and Presnell (2017) interpret universes "by understanding them as domains of discourse, where a domain of discourse consists of the concepts and propositions that are understood and defined in a given discussion" ( §2.3) and observe that this characterization can justify the usual rules for universes (UFP, 2013, § A.1.1-8,A.2.3-10). Such a detailed discussion would take us too far from our subject, but the crucial point here is the extensional character that is attributed to domains of discourse (it is essential that types have an extensional criterion of identity for the justification of univalence, otherwise we could not identify them when they are equivalent).…”

“…(Ladyman and Presnell, 2016, §5.2.4) Extensional entities should always be compatible with intensional criteria of identity, but as far as intensional entities and extensional equality are concerned this statement is clearly false. It is natural to expect an explanation of why domains of discourse (which are general concepts) contradict the inherent intensionally of concepts, but no such clarification is given in their account (Ladyman and Presnell, 2017).…”