Univalent foundations and the equivalence principle

TL;DR

This paper discusses how univalent foundations support the equivalence principle, ensuring mathematical statements are invariant under appropriate equivalences, contrasting with set-theoretic foundations where this invariance often fails.

Contribution

It demonstrates how univalent foundations validate the equivalence principle for various mathematical structures, highlighting its advantages over traditional set-theoretic approaches.

Findings

Univalent foundations validate the equivalence principle for many structures.

Set-theoretic foundations often fail to uphold the equivalence principle.

The paper provides an overview of foundations satisfying the equivalence principle.

Abstract

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