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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.
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Applications of opetopes
I've been reading about coherence problems in homotopy type theory (regarding semisimplicial sets and a raw syntax interpreter), and I've seen a remark about higher-dimensional operads perhaps being t …