Homotopy Type Theory

Real-cohesive homotopy type theory

Two new papers have recently appeared online:

Both of them have fairly chatty introductions, so I’ll try to restrain myself from pontificating at length here about their contents. Just go read the introductions. Instead I’ll say a few words about how these papers came about and how they are related to each other.

Cohesive homotopy type theory is something that I’ve been working on with Urs Schreiber since some 3-4 years ago, involving an adjoint string of modalities ʃ that internalizes Lawvere’s axiomatic cohesion. We wrote a little note about it. About a year ago, I started to wonder whether cohesive HoTT could solve “the problem of the two circles”. By this I mean the fact that in HoTT we have both the higher inductive circle, generated by a point base and an equality loop : base = base, and the topological circle, defined for instance as . The former is a 1-type that has “only one point” (to be precise, it is 0-connected), whereas the latter is a 0-type (a set) that has infinitely many distinct points. The “problem” is, how are these two circles related?

In March I gave a talk about this at CMU, in which I introduced an enhancement of cohesive HoTT called “real-cohesive HoTT” and proved in it that the “shape” ʃ (the cohesive incarnation of the fundamental -groupoid) of the topological circle is the homotopical circle. At that point Dan Licata, who was in the audience, got interested and started convincing me to look for a better way to formulate cohesion type-theoretically.

The problem with cohesive type theory is that the comodality , unlike the modalities ʃ and , can’t be described internally as a map with properties. Urs and I first noticed this semantically; later I proved a no-go theorem internally (see section 4 of the real-cohesion paper). Urs and I worked around this by representing as a map ; while later, I started experimenting with using Coq’s modules to restrict the context. But in March Dan told me about modal type theories due to Pfenning-Davies and Reed, in which a segregation of the context limits the applicability of modal type formers. The existing work had focused on more traditional modalities such as necessity and possibility , but Dan thought there ought to be a version that would work for the cohesive modalities as well.

There followed a very productive back-and-forth in which Dan developed the metatheory with an eye towards a proof-theoretically well-behaved calculus, while I cast around for a version of the theory that would make for a perspicuous presentation of real-cohesion. We ended up with an idea for a very general kind of “adjoint type theory”, which is not yet worked out in full; the real-cohesion paper uses only one particular case of it, while the adjoint logic paper develops the metatheory of the general case but in a very simplified way (no dependent types and only one-variable contexts).

Dan coded up a simple sequent calculus for this theory in Agda and proved some nice theorems about it. Meanwhile I generalized it to dependent type theory in the special case of cohesion and used it to write up the two circles theorem, and also a synthetic version of the classical homotopy-theoretic proof of Brouwer’s fixed-point theorem as a proof of concept. I’m very excited about real-cohesion, and also adjoint type theory more generally; I think it has a lot of untapped potential. A few possibilities are mentioned in the paper.

Unfortunately there is one fly in the ointment: cohesive type theory done in this way can’t be formalized directly in existing proof assistants. One can describe the theory using a proof assistant as the metatheory, of course.  For actually proving things in the theory, this works pretty well for basic theorems in a 1-variable sequent calculus (see the Agda code accompanying the adjoint logic paper), but for more complicated results in a richer language it would become quite tedious, running into the common problems with representing languages with variable binding and dependent types. At one point I entertained notions that Coq’s modules would help; but at the moment I’m kind of down on that idea because modules have a lot of “features” that make them at best annoying to use in this way (which, to be fair, is not at all how they were intended to be used). So formalization may have to wait until someone implements an adjoint-type-theory proof assistant directly. (Anyone developing a proof assistant feel like incorporating it?)