The "equivalence principle" (EP) says that meaningful statements in
mathematics should be invariant under the appropriate notion of
equivalence - "sameness" - of the objects under consideration.
In set theoretic foundations, the EP is not enforced; e.g., the
statement "1 ϵ Nat" is not invariant under isomorphism of sets.
In univalent foundations, on the other hand, the equivalence principle
has been proved for many mathematical structures.
In this introductory talk, I first give an overview of earlier attempts
at designing foundations that satisfy some invariance property.
I will describe a geometric chain level construction of a secondary coproduct operation on a suitable chain model for the free loop space of a manifold using the theory of De Rham chains. Such coproduct was described at the level of homology by Goresky and Higston using different methods. It is secondary in the sense that arises from a "1-parameter family of chain level intersections". The chain level theory around this secondary operation is useful for describing certain phenomena in symplectic topology and symplectic field theory.