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In this paper, we explore the 'equivalence principle' (EP): roughly,statements about mathematical objects should be invariant under an appropriatenotion of equivalence for the kinds of objects under consideration. In settheoretic foundations, EP may not always hold: for instance, the statement '1\in N' is not invariant under isomorphism of sets. In univalent foundations, onthe other hand, EP has been proven for many mathematical structures. We firstgive an overview of earlier attempts at designing foundations that satisfy EP.We then describe how univalent foundations validates EP.

Univalent foundations and the equivalence principle