Discussing "The Simplicial Model of Univalent Foundations (after Voevodsky)"

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Authors: Krzysztof Kapulkin (University of Pittsburgh) and Peter LeFanu Lumsdaine (Institute for Advanced Study)

Reference: Krzysztof Kapulkin, Peter LeFanu Lumsdaine, The simplicial model of Univalent Foundations (after Voevodsky). J. Eur. Math. Soc. 23 (2021), no. 6, pp. 2071–2126. DOI 10.4171/JEMS/1050

Why this paper? Cited by Normalization and coherence for ∞-type theories, The Formal Theory of Monads, Univalently, and On the ∞-topos semantics of homotopy type theory

This paper has a rather peculiar publication history, as it started as a 2012 preprint written by the two authors along with a third, Fields medal winning mathematician Vladimir Voevodsky, who was in fact the main author. However Voevodsky grew disenchanted with the paper due to some issues with the type theory literature it was relying on, and asked for his name to be removed. Sadly, Voevodsky passed away before rectifying these holes in the literature, and so Kapulkin and Lumsdaine worked the paper into a publication. Lacking permission to re-add Voevodsky’s name to the author list, his contribution is instead recognised in the title.

One benefit of this paper having almost a decade to germinate is that it has a particularly pleasant and well written introduction to its topic, Homotopy Type Theory (HoTT). I looked at the seminal HoTT book on this blog last year; this paper picks up one of the topics that the book does not address, although it is briefly mentioned, namely denotational semantics. To recap, HoTT is an approach to mathematical foundations that aims to be both usable by humans and amenable to computer formalisation. The technical approach is to extend previous approaches to type theory with, in particular, a property called the univalence axiom (the other major innovation, higher inductive types, are not discussed here).  Univalence strengthens the usual notion of equality in type theory to, at the type level, correspond to equivalence; this means that one can move constructions and results between equivalent mathematical settings, a common feature of mathematical argument, ‘for free’. This leads one to think of types as (some sort of topological) space, with equality as paths in a space à la homotopy theory.

This paper makes this connection formal by giving homotopy type theory denotational semantics in the category of simplicial sets. Denotational semantics give a mathematical meaning for the constructions of a type theory, programming language, or logic. So long as the mathematical model chosen is not pathological, this gives us as immediate benefit a proof of consistency (one cannot prove false, or construct an element of a type that should be empty). Just as importantly, this style of semantics assures us that all syntactic constructions are not mere shuffling of symbols, but have real mathematical meaning that can be communicated to someone who is not necessarily interested in the details of the particular type theory. Put another way, the type theory is ‘about something’ other than itself.

Why is the ‘something’ that HoTT is about in this paper – i.e. simplicial sets – actually interesting, beyond mere proof of consistency? This paper does not offer an introduction to this mathematics, but looking at one of the books cited as background (link to pdf) gives some clues: “the category of simplicial sets has a homotopical structure … the resulting homotopy theory is equivalent in a strong sense to the ordinary homotopy theory of topological spaces… Insofar as simplicial sets are algebraically defined, and the corresponding closed model structure is combinatorial in nature, we obtain an algebraic, combinatorial model for standard homotopy theory”. In other words, simplicial sets are not a mere mathematical curiosity than happens to work for this type theory, but rather a strongly motivated approach to the general study of homotopy which, of course, is what motivated homotopy type theory in the first place. It should be noted that simplicial sets are not unique in having this property; there is a somewhat similar structure called cubical sets that appear to be more suited to computational implementation. But that, perhaps, is a story for another day.