Title of the course: Category theory in geometry and physics
Instructor: Dr. Kadri ilker Berktav
Institution: University of Zurich
Dates: 4-10 September 2023
Prerequisites: Basic knowledge of algebra, algebraic geometry, and physics
Level: Graduate, advanced undergraduate
Abstract: This is an introductory course on (higher) category theory and its possible roles in geometry and physics. The course will be self-contained, but some background from algebra, algebraic geometry/topology, and a little bit of physics would be helpful to better understand the examples. In the first part of the course, we present the basics of category theory along with some examples. In the following part, we discuss (very briefly and informally) higher categories, for which we only mention some guiding principles and key ideas.  Having established the language we need, we then provide our first big example regarding the roles of categorical constructions: Yoneda’s lemma, which eventually leads to defining geometric objects (like schemes, varieties, or manifolds) as certain functors. With this functorial perspective, we also briefly discuss how to define the so-called “higher” spaces (like stacks, derived schemes/stacks) and then tease a general framework for these objects (a.k.a. derived algebraic geometry). In the final part of this course, we give the second example of categorical constructions: Functorial field theories. In this part, we explain how to formulate a physical theory using a particular functor between some (higher) categories.
Language: EN