Discussing "Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom"

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Authors: Cyril Cohen (INRIA Sophia Antipolis - Méditerranée), Thierry Coquand (University of Gothenburg), Simon Huber (University of Gothenburg), and Anders Mörtberg (Institute for Advanced Study)

Reference: Cyril Cohen, Thierry Coquand, Simon Huber, Anders Mörtberg, Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom. In 21st International Conference on Types for Proofs and Programs (TYPES 2015), ed. Tarmo Uustalu, Leibniz International Proceedings in Informatics (LIPIcs), vol. 69 (2018).

Why this paper? Cited by Formalizing CCS and π-Calculus in Guarded Cubical Agda and Normalization and coherence for ∞-type theories

I’ve blogged a few times on homotopy type theory, ranging from the seminal book to various follow-ups (One Two Three). This is one of the most important follow-ups because of what it does to the univalence axiom, the cornerstone of homotopy type theory that asserts that equivalences between types correspond to identities between them. Cubical type theory is an instance of homotopy type theory for which the univalence axiom is not an axiom at all, but rather a provable theorem (inhabited type). The significance of this may not be immediately obvious to newcomers to type theory – who cares if a statement is an axiom or a consequence of some other rules? In fact with the type theories that lend themselves to applications such as interactive theorem proving we do not typically proceed by asserting axioms – unless we need to – but rather by defining rules for introducing and eliminating our basic type formers. Arranging things in this way allow us to have good computational properties, for example a constructive argument involving a natural number really will produce an appropriate natural number when run. This gives proofs in the type theory computational content, whereas the bald assertion of an axiom would provide a ‘roadblock’ in the execution of a proof.

This development proceeds by developing a type theory which has denotational semantics in the category of cubical sets. In this blog post I won’t sweat the details of these semantics – many of my comments last week about semantics in simplicial sets still apply  – although it is important to understand that working via the semantics motivates every part of this paper. However, I wish to concentrate on the details of the type theory itself as syntax.

It is fair to say that cubical type theory is a heavy duty extension to standard type theory. Some of the earlier parts of the paper are quite beautiful, however. The notion of identity as path is captured by thinking of them as (roughly) a function from a slightly abstracted version of the real interval [0,1] to a type; if we wish to express that t is equal u then we need that function to map 0 to t and 1 to u, and we’re done. This gives a startlingly simple proof of function extensionality (functions are equal if they agree on all inputs), which is a tricky property, and indeed sometimes not even true, in many type theories. Things get more involved, but still rather elegant, if we want dependence on multiple different arbitrary elements of the interval, usually called dimensions. This is where the cubes of cubical type theory come in, as zero dimensions give rise to a point in a space (which is to say, just a usual inhabitant of a type), and one dimension to a path in a space, but we can have squares, cubes, 4-dimensional hypercubes and so on as well.

The next feature is to define faces, which can be thought of as corresponding to faces of cubes (of various dimension), or more prosaically as restrictions on the values that our otherwise arbitrary interval elements can take. For example, we might require that both i and j are equal to one of 0 or 1, describing the outline of a square but leaving its centre unfilled. Terms and types defined with faces in their context are then only partial, e.g. only existing where the free dimensions i and j have appropriate values. Systems are terms of arbitrary type that explicitly define different constructions for different faces. Faces are also used to introduce the notion of extensibility;  a partial term t with respect to face φ is extensible if there is a term u that is not restricted to φ, for which if φ is added to its context, it is indeed equal to t. A new composition operator then comes into play that ensures that extensibility is preserved along paths. This may seem heavy, but is necessary to prove the rather mundane fact that identity is transitive. The equality judgements (intuitively, reduction rules) associated with the composition operator are fairly complex, depending on the type the composition is applied at and so needing explicit definition corresponding to each type former.

The final layer of novelty involves a new type former called glueing which connects types with partial types, and ensures that extensibility is invariant across equivalence. This is the final necessary ingredient for univalence (which is a statement about equivalences, after all) to be proved.