# Poly @ Work 2024

Topos Institute, February 19 – March 1

## Motivation

Polynomial functors are a prime example of mathematics’ startling capacity to yield deep formal insights and real-world applications from the simplest ingredients. The category of polynomial functors (Poly) offers an abundance of elegant theory linked with computational design patterns that span the entire abstraction hierarchy. It is a rich playground in which ideas from different disciplines naturally collide.

This workshop brought together students, researchers, and practitioners from around the world to learn, share, and collaborate on the theory and applications of polynomial functors.

The workshop was organized by Sophie Libkind, Shaowei Lin, Nate Osgood, and David Spivak.

## Participants

The participants had a range of expertises and interests including:

- Neuroscience, cognitive science, and psychology
- Health and Epidemiology
- Computer Science and Software Engineering
- Mathematics
- Philosophy of science

The full list of participants is as follows:

- Solomon Bothwell
- Matteo Capucci
- Max Collard
- Peter Gates
- Harrison Grodin
- Xiaoyan Li
- Sophie Libkind
- Joshua Meyers
- Reed Mullanix
- André Muricy
- Nelson Niu
- Nate Osgood
- Thomas Purdy
- Eric Redekopp
- Brandon Shapiro
- Toby St Clere Smithe
- David Spivak
- Priyaa Srinivasan
- Lauresa Stilling

## Program

The workshop ran for two weeks and was hosted at the Topos Institute in Berkeley, California.

### Learning week

In the first week, participants learned about the basics of polynomial functors through lectures by David Spivak, exercises, and conversations. Videos are available on Youtube; see the resources section below.

Topics included:

**Theory**

- The category of polynomial functors in one variable
- Four monoidal products
- Two monoidal closures
- Monoids and comonoids
- Bicomodules

**Applications**

- Dynamical systems (coalgebras)
- Wiring diagrams (polynomial maps)
- Databases and data migration (bicomodules)

### Research week

In the second week, participants explored Poly-shaped problems using the language of polynomial functors. Each participant produced a written artifact of their research, which can be found in the resources section below.

## Resources

### YouTube lectures

All the lecture videos can be found in the playlist Polynomial functor lecture series on the Topos Institute channel.

### Artifacts

The following artifacts were produced in 90 minutes on the final day of Research week. They are snapshots of the participants’ research at the time and should not be treated as finished products. They may also contain typos.

- Pairing dynamic systems (Solomon Bothwell)
- A glimpse of the algebraic theory of linear systems (Matteo Capucci)
- Changing minds (Max Collard)
- Some Takeaways from Poly at Work 2024 (Harrison Grodin)
- An idea of creating (compositional) particle filtering dynamical models in Poly (Xiaoyan Li)
- Three views on Org (Sophie Libkind)
- Sketching the elephant: 7 perspectives on Poly (Reed Mullanix)
- Bicomodules for Poly-shaped reservoir computer (André Muricy Santos)
- Polynomials in Action (Nelson Niu)
- PolyMapBasic.jl (Thomas Purdy)
- Particle filtering with polynomial functors (Eric Redekopp)
- The Poly-shaped ingredients of predictive coding (Toby St Clere Smithe)
- Applications of multivariate polynomials (David Spivak)
- How to win a lottery? (Priyaa Varshinee Srinivasan)
- Exploring Poly through Catlab’s Poly.jl (Lauresa Stilling)
- Notes from the workshop (Nate Osgood)