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 like limits, adjunctions, monoidal products, closures, monoids, modules, and bimodules. 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.

‘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 Source: 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