This webpage 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
Polynomial Functors: A General Theory of Interaction
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 and future livestreams can be found here.
- YouTube Day 1: Introduction §§ 1.1–1.3 Thu, Jul 15
- YouTube Day 2: Polynomial morphisms §§ 1.3–1.4, 2.2–2.4 Mon, Jul 19
- YouTube Day 3: Dynamics of polynomials I §§ 3.1–3.2 Thu, Jul 22
- YouTube Day 4: Dynamics of polynomials II §§ 3.3–3.4 Mon, Jul 26
- YouTube Day 4.5: The double category of arenas Wed, Jul 28 (special guest lecture by David Jaz Myers)
- YouTube Day 5: Categorical properties of polynomials §§ 3.5, 4.1–4.4 Thu, Jul 29
- YouTube Day 6: The composition product §§ 5.1–5.4 Mon, Aug 2
- YouTube Day 6.5: Behaviors compose by matrix arithmetic Wed, Aug 4 (special guest lecture by David Jaz Myers)
- YouTube Day 7: Polynomial comonoids are categories §§ 6.1–6.4 Thu, Aug 5
- YouTube Day 8: Categories and cofunctors §§ 6.5, 6.7 Mon, Aug 9
- YouTube Day 9: Cofree polynomial comonoids §§ 7.1–7.7 Thu, Aug 12
- YouTube Day 10: Products of polynomial comonoids § 6.6 Mon, Aug 16
- YouTube Day 11: Bimodules over polynomial comonoids… § 9.1 Thu, Aug 19
- YouTube Day 12: …are parametric right adjoints § 9.2 Mon, Aug 23
- YouTube 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
- the Yoneda lemma
- monoidal categories, including those that are symmetric and/or closed
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-20-1-0348. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Air Force.