Polynomial Functors: A Mathematical Theory of Interaction(Series Number 498 London Mathematical Society Lecture Note Series)

Everywhere one looks, one finds dynamic interacting systems: entities expressing and receiving signals between each other and acting and evolving accordingly over time. In this book, the authors give a new syntax for modeling such systems, describing a mathematical theory of interfaces and the way they connect. The discussion is guided by a rich mathematical structure called the category of polynomial functors. The authors synthesize current knowledge to provide a grounded introduction to the material, starting with set theory and building up to specific cases of category-theoretic concepts such as limits, adjunctions, monoidal products, closures, comonoids, comodules, and bicomodules. The text interleaves rigorous mathematical theory with concrete applications, providing detailed examples illustrated with graphical notation as well as exercises with solutions. Graduate students and scholars from a diverse array of backgrounds will appreciate this common language by which to study interactive systems categorically.

Table of Contents:
Part I. The Category of Polynomial Functors: 1. Representable functors from the category of sets; 2. Polynomial functors; 3. The category of polynomial functors; 4. Dynamical systems as dependent lenses; 5. More categorical properties of polynomials; Part II. A Different Category of Categories: 6. The composition product; 7. Polynomial comonoids and retrofunctors; 8. Categorical properties of polynomial comonoids; References; Index.

About the Author :
Nelson Niu is a Ph.D. Student in the Department of Mathematics at the University of Washington. He was a keynote speaker on Polynomial Functors at the 2022 Artificial General Intelligence Conference. He conducted research in applied category theory with David I. Spivak at MIT and currently consults with NASA on category theory applied to Advanced Air Mobility Architectures. David I. Spivak is Senior Scientist and Institute Fellow at Topos Institute. He earned his Ph.D. in mathematics from UC Berkeley in 2007. He went on to demonstrate the broad applicability of category theory during his postdoctoral work and ten years at MIT. He also co-founded the Topos Institute and has authored three books on category theory applications.

Review :
'Crafted with evident care for the subject and the reader, Niu and Spivak invite us into the mathematically abundant world of polynomial functors. Their practical and pedagogical approach plants the seeds for a long, fruitful interaction between 'Poly' and those making sense of our dynamic and interconnected world.' Brendan Fong, Topos Institute 'This book by Niu and Spivak is a new perspective on automata and dynamical systems. It contains new kinds of mathematics, but is fun and easy to read. It is all about polynomials, but of a new kind. It is all about lenses, but what is a lens? Do you know that a Moore machine is a special kind of lens? That a polynomial comonoid is the same thing as a category? Do you know what a retrofunctor is? The book answers all these questions and more. It offers plenty of solved exercises.' André Joyal, Université du Québec à Montréal 'In this lovingly illustrated volume, Niu and Spivak gift the reader an admirably accessible treasure trove offering profound value for addressing challenges in diverse application areas, including databases, dynamical systems, simulation, programming language semantics and type theory. It is hard to overstate the contribution of this book, so full of wonderfully explicated concepts, insightful examples, and thought-provoking exercises that build readers' capacity to actualize the potential of polynomial functors in their own spheres of interest.' Nathaniel Osgood, Computational Epidemiology & Public Health Informatics Laboratory, University of Saskatchewan