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5913 results in Programming Languages and Applied Logic

Page 5 of 296

  • 2 - Polynomial functors

  • from Part I - The category of polynomial functors
  • Nelson Niu, University of Washington, David I. Spivak, Topos Institute
  • Book:
    Polynomial Functors
    Published online:
    27 September 2025
    Print publication:
    16 October 2025, pp 29-50

  • Summary

    We formally define polynomial endofunctors on the category of sets, referring to them as polynomial functors or simply polynomials. These are constructed as sums of representable functors on the category of sets. We provide concrete examples of polynomials and highlight that the set of representable summands of a polynomial is isomorphic to the set obtained by evaluating the functor at the singleton set, which we term the positions of the polynomial. For each position, the elements of the representing set of the corresponding representable summand are called the directions. Beyond representables, we define three additional special classes of polynomials: constants, linear polynomials, and monomials. We close the chapter by offering three intuitive interpretations of positions and directions: as menus and options available to a decision-making agent, as roots and leaves of specific directed graphs called corolla forests, and as entries in two-cell spreadsheets we refer to as polyboxes.

  • 7 - Polynomial comonoids and retrofunctors

  • from Part II - A different category of categories
  • Nelson Niu, University of Washington, David I. Spivak, Topos Institute
  • Book:
    Polynomial Functors
    Published online:
    27 September 2025
    Print publication:
    16 October 2025, pp 295-376

  • Summary

    We show that the category of comonoids, defined with respect to the composition product in the category of polynomial functors, is equivalent to a category of small categories as objects but with an interesting type of morphism called retrofunctors. Unlike traditional functors, retrofunctors operate in a forward-backward manner, offering a different kind of relationship between categories. We introduce this concept of retrofunctors, provide examples to illustrate their behavior, and explain their role in modeling state systems.

  • 4 - Dynamical systems as dependent lenses

  • from Part I - The category of polynomial functors
  • Nelson Niu, University of Washington, David I. Spivak, Topos Institute
  • Book:
    Polynomial Functors
    Published online:
    27 September 2025
    Print publication:
    16 October 2025, pp 105-184

  • Summary

    We model discrete-time dynamical systems using a specific class of lenses between polynomials whose domains are equipped with a bijection between their positions and their directions. We introduce Moore machines and deterministic state automata as key examples, showing how these morphisms describe state transitions and interactions. We also explain how to build new dynamical systems from existing ones using operations like products, parallel composition, and compositions of these maps. This chapter demonstrates how polynomial functors can be used to represent and analyze discrete-time dynamical behavior in a clear, structured way.

  • 5 - More categorical properties of polynomials

  • from Part I - The category of polynomial functors
  • Nelson Niu, University of Washington, David I. Spivak, Topos Institute
  • Book:
    Polynomial Functors
    Published online:
    27 September 2025
    Print publication:
    16 October 2025, pp 185-230

  • Summary

    We describe a range of additional category-theoretic structures on the category of polynomial functors. These include concepts like adjunctions, epi-mono factorizations, and cartesian closure. We also cover limits and colimits of polynomials and explore vertical-cartesian factorizations of morphisms. The chapter highlights how these structures provide new tools for working with polynomial functors and extend their usefulness in modeling various types of interactions and constructions.

Polynomial Functors
  • A Mathematical Theory of Interaction
  • Nelson Niu, David I. Spivak
  • Published online:
    27 September 2025
    Print publication:
    16 October 2025
  • 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.

Page 5 of 296