Abstract

In this talk I will present an introduction to dependent type theory, covering dependent pairs and functions, identity types and universes. I will talk about the use of type theory as a constructive foundation for mathematics and give some examples of encoding proofs using dependent types. I will then introduce the univalence axiom, which implies that equivalent types are equal. I will show how univalence is incompatible with the principle that all proofs of equality are the same (uniqueness of identity proofs) and how other principles, such as function extensionally, follow from the univalence axiom. If time allows I will also cover other related topics, such as higher inductive types, synthetic homotopy theory or cubical type theory.