Univalence as a principle of logic

Abstract

It is sometimes convenient or useful in mathematics to treat isomorphic structures as the same. The recently proposed Univalence Axiom for the foundations of mathematics elevates this idea to a foundational principle in the setting of homotopy type theory. It provides a simple and precise way in which isomorphic structures can be identified. We explore the motivations and consequences, both mathematical and philosophical, of making such a new logical postulate.