Polynomial Functors: A General Theory of Interaction
15 July – 26 August 2021
This page is for a course that was run in 2021. Although the course is now finished, recordings of the lectures can be found below, along with a link to the book.
About
The category of polynomial functors is a fascinating setting, brimming with rich mathematics and tantalizing applications. In this course, we will investigate these polynomials, with emphasis on their applications to dynamical systems, decisions, and data. We aim to strike a balance between a solid theoretical foundation and a breadth of examples.
Instructors: Nelson Niu & David Spivak
Book
A work-in-progress! Feedback is welcome and may be shared here. Class discussions will likely inform how we present this content within the book and in future work.
Recordings
The YouTube playlist of all recorded lectures can be found here.
Day 1: Introduction (§§ 1.1–1.3). Thu, Jul 15.
Day 2: Polynomial morphisms (§§ 1.3–1.4, 2.2–2.4). Mon, Jul 19.
Day 3: Dynamics of polynomials I (§§ 3.1–3.2). Thu, Jul 22.
Day 4: Dynamics of polynomials II (§§ 3.3–3.4). Mon, Jul 26.
Day 4.5: The double category of arenas Wed, Jul 28 (special guest lecture by David Jaz Myers).
Day 5: Categorical properties of polynomials (§§ 3.5, 4.1–4.4). Thu, Jul 29.
Day 6: The composition product (§§ 5.1–5.4). Mon, Aug 2.
Day 6.5: Behaviors compose by matrix arithmetic Wed, Aug 4 (special guest lecture by David Jaz Myers).
Day 7: Polynomial comonoids are categories (§§ 6.1–6.4). Thu, Aug 5.
Day 8: Categories and cofunctors (§§ 6.5, 6.7). Mon, Aug 9.
Day 9: Cofree polynomial comonoids (§§ 7.1–7.7). Thu, Aug 12.
Day 10: Products of polynomial comonoids (§ 6.6). Mon, Aug 16.
Day 11: Bimodules over polynomial comonoids… (§ 9.1). Thu, Aug 19.
Day 12: …are parametric right adjoints (§ 9.2). Mon, Aug 23.
Day 13: Bimodules and further discussion (§§ 9.3–9.4, 10.1–10.3). Thu, Aug 26.
Prerequisites
You should be able to define the following fundamental concepts from category theory:
- categories
- functors
- natural transformations
- (co)limits
- adjunctions
We’ll discuss the following as if you’ve seen them before but may need a quick refresher:
- the Yoneda lemma
- monoidal categories, including those that are symmetric and/or closed