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.
Spivak / Niu
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Weitere Infos & Material
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; 9. Future work in polynomial functors; References; Index.
Niu, Nelson
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.
Spivak, David I.
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.
