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The homotopy interpretation of constructive dependent type theory, the univalence axiom, higher inductive types, internal languages of higher toposes, univalent foundations for mathematics, and implementations of such theories in proof assistants.
8
votes
How do you define (infinity,1) categories in Homotopy Type Theory?
While this question has an accepted answer, let me just mention that there is now a preprint by James Cranch about doing categories structured over homotopy types, you might be interested to have a lo …