We define the category of polynomial functors by introducing its morphisms, called dependent lenses or lenses for short, and we show how they model interaction protocols. We introduce several methods for working with these lenses, including visual tools such as corolla forests and polybox pictures. We explain how these lenses represent bidirectional communication between polynomials and describe how they compose. By the end of the chapter, readers will have a comprehensive understanding of how polynomial functors and their morphisms can be used to model complex interactive behaviors.

We review relevant concepts from and properties of the categories of sets and of endofunctors on the category of sets relevant to our work. We discuss representable functors on the category of sets, introducing our exponential notation for them, and we state and prove the Yoneda lemma for these with the help of an exercise. We then examine sums (or coproducts) and products of sets and functions through the language of indexed families of sets. In particular, we characterize products of sets in terms of dependent functions, generalizing functions by allowing their codomains to vary depending on their inputs. We study nested sums and products of sets, explaining how distributivity allows us to expand products of sums of sets. By lifting all of this material to endofunctors on the category of sets, and using the fact that its limits and colimits are computed pointwise, we set ourselves up to introduce polynomial functors as sums of representable functors in the next chapter. Throughout the chapter, we emphasize key categorical principles and provide detailed explanations to ensure solid comprehension of these fundamental ideas.

We examine a monoidal structure on the category of polynomial functors, defined through the operation of substituting one polynomial into another. We explain how this composition product transforms polynomials into a richer algebraic structure, enabling the modeling of more complex interactions and processes. The chapter explores the properties of this monoidal structure, how it relates to existing constructions in category theory, and its implications for understanding time evolution and dynamical behavior. We also provide examples and visual representations to clarify how substitution works in practice.

We study the structure and utility of the category formed by small categories and retrofunctors. We analyze key properties of this category, such as limits, colimits, and factorizations, and explain how these structures support various forms of composition and interaction. The chapter delves into the cofree comonoid construction, exploring how it connects to familiar concepts in category theory, and extends our understanding of state-based systems. We also discuss applications of retrofunctors and demonstrate how they can be used to model complex processes in a structured way.

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.

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.

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.

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.