Abstraction, equality and univalence.

We shall propose a conceptual-oriented discussion of the so-called Univalent Foundations Program, that is, of Martin-Löf type theory enriched with a homotopic interpretation, together with the univalence axiom proposed by Voevodsky. We shall argue that the type-theoretic notion of encodes the notion of , we shall address the homotopic interpretation of Martin-Löf type theory, and we shall analyse whether Leibniz's principle of the identity of indiscernibles holds or not in Univalent Foundations. We shall finally argue that univalence can be understood as a particular implementation of a constructive notion of abstraction that resolves Fregean abstraction. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.