Discussion of Homotopy Type Theory and Univalent Foundations
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* [HoTT] CMU HoTT Seminar Online: M. Shulman, Towards Third-Generation HOTT, April 28, May 5 & 12
@ 2022-04-22 20:44 Steve Awodey
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From: Steve Awodey @ 2022-04-22 20:44 UTC (permalink / raw)
  To: Homotopy Type Theory, CMU HoTT
  Cc: EMILY RIEHL, Michael Shulman, Egbert😁 Rijke, Steve Awodey

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

Mike Shulman (University of San Diego).
April 21, 28 and May 5
11:30am-1:00pm EST (UTC-04:00)

Join Zoom Meeting

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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