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Condensed Type Theory

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If you have a question about this talk, please contact Jonas Bayer.

Condensed sets form a topos, and hence admit an internal type theory. In this talk I will describe a list of axioms satisfied by this particular type theory. In particular, we will see two predicates on types, that single out a class CHaus of “compact Hausdorff” types and a class ODisc of “overt and discrete” types, respectively. A handful of axioms describe how these classes interact. The resulting type theory is spiritually related Taylor’s “Abstract Stone Duality”.

As an application I will explain that ODisc is naturally a category, and furthermore, every function ODisc → ODisc is automatically functorial. This axiomatic approach to condensed sets, including the functoriality result, are formalized in Lean 4. If time permits, I will comment on some of the techniques that go into the proof.

Joint work with Reid Barton.

This talk is part of the Formalisation of mathematics with interactive theorem provers series.

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