CAUTION: The Sender of this email is not from within Dalhousie. [Note from moderator: Since I will be mostly without suitable internet access for several days, list service may be unavailable until May 11.] Reminder: Part II, May 5. ~~~~~~~~ CMU HoTT Seminar Online ~~~~~~~~ Mike Shulman (University of San Diego). April 28, May 5, May 12 11:30am-1:00pm EST (UTC-04:00) Join Zoom Meeting https://cmu.zoom.us/j/622894049 Meeting ID: 622 894 049 Passcode: Brunerie's number Mike Shulman University of San Diego Towards Third-Generation HOTT

On Apr 23, 2022, at 14:15, Steve Awodey <awodey@cmu.edu> wrote:

~*~*~*~*~*~*~*~* CMU HoTT Seminar Online *~*~*~*~*~*~*~*~

Mike Shulman (University of San Diego). April 28, May 5, May 12 11:30am-1:00pm EST (UTC-04:00)

Join Zoom Meeting https://cmu.zoom.us/j/622894049

Meeting ID: 622 894 049 Passcode: the Brunerie number

************************************************************

Mike Shulman University of San Diego

Towards Third-Generation HOTT

In Book HoTT, identity is defined uniformly by the principle of "indiscernibility of identicals". This automatically gives rise to higher structure; but many desired equalities are not definitional, and univalence must be asserted by a non-computational axiom. Cubical type theories also define identity uniformly, but using paths instead. This makes more equalities definitional, and enables a form of univalence that computes; but requires inserting all the higher structure by hand with Kan operations.

I will present work in progress towards a third kind of homotopy type theory, which we call Higher Observational Type Theory (HOTT). In this system, identity is not defined uniformly across all types, but recursively for each type former: identifications of pairs are pairs of identifications, identifications of functions are pointwise identifications, and so on. Univalence is then just the instance of this principle for the universe. The resulting theory has many useful definitional equalities like cubical type theories, but also gives rise to higher structure automatically like Book HoTT. Also like Book HoTT, it can be interpreted in a class of model categories that suffice to present all Grothendieck-Lurie (∞,1)-toposes; and we have high hopes that, like cubical type theories, some version of it will satisfy canonicity and normalization.

This is joint work with Thorsten Altenkirch and Ambrus Kaposi.

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