An introduction to homotopy type theory and Univalent Foundations

14 May 2014
10:30
Emily Cliff
Abstract
In this talk we aim to introduce the key ideas of homotopy type theory and show how it draws on and has applications to the areas of logic, higher category theory, and homotopy theory. We will discuss how types can be viewed both as propositions (statements about mathematics) as well as spaces (mathematical objects themselves). In particular we will define identity types and explore their groupoid-like structure; we will also discuss the notion of equivalence of types, introduce the Univalence Axiom, and consider some of its implications. Time permitting, we will discuss inductive types and show how they can be used to define types corresponding to specific topological spaces (e.g. spheres or more generally CW complexes).\\ This talk will assume no prior knowledge of type theory; however, some very basic background in category theory (e.g. the definition of a category) and homotopy theory (e.g. the definition of a homotopy) will be assumed.