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Type theoretic aspects of AU sketches

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If you have a question about this talk, please contact Dr Steve Vickers.

Apologies for wrong announcement that it was Wednesday

This continues a talk I started some weeks ago.

In my paper “Sketches for arithmetic universes” I define a 2-category Con in which each object is intended to be a context in a “dependent type theory of spaces”. I’ll discuss two different ways in which type theoretic ideas can be brought in.

On the one hand, each context T generates a “dependent type theory of sets” which, following Maietti’s work, will be a language for the classifying AU of T.

On the other hand, the system has the structure of category with attributes and that allows us to relate different contexts. A “type” in a context T is then one of the different possible sets of “extension data” for that context, and shows how to extend T. Then the terms are found using “equivalence extensions”, and that takes us back to the other kind of type theory.

I may also mention some of my experiments with implementing this in Agda.

This talk is part of the Lab Lunch series.

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