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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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Base change in homotopy type theory
Recall that with the internal language of 1-toposes, we have the nice, basic, and useful result that geometric sequents are stable under base change along geometric morphisms: If $\varphi$ and $\psi$ …