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On Cubes and Types

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If you have a question about this talk, please contact Neel Krishnaswami.

(joint work with Ambrus Kaposi)

Homotopy Type Theory (HoTT) extends Martin-Loef Type Theory with the univalence principle, a strong extensionality principle that internalizes the idea that equivalent structures should be considered equal. At the same time we are lead to view types as higher dimensional objects in the sense of homotopy theory.

Due to its extensional nature HoTT seems ideally suited to support mathematical abstraction and may hence be indispensable when building large libraries of formalized mathematics and verified programs.

Currently, HoTT faces one major obstacle: we do not know how to extend the computational principles from conventional Type Theory to HoTT. To address this issue we propose a different way to introduce equality types in Type Theory as an heterogenous equality defined as a logical relation. This approach is clearly related to Bernardy’s and Moulin’s internal parametricity and Coquand’s and Huber’s cubical set model. Due to the presence of higher dimensional cubes in all these approaches we call this Cubical Type Theory. In the talk I will give a gentle introduction to cubical type theory and try to explain how far we got and what remains to be done.

This talk is part of the Theoretical computer science seminar series.

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