Polynomial Functors: A Mathematical Theory of Interaction — Nelson Niu & David I. Spivak (2023). arXiv:2312.00990 (v2, PDF).
A book-length treatment of the category of polynomial functors on Set and dependent lenses between them, as a theory of interactive systems: positions are what a system shows, directions what it accepts. Covers the monoidal structures of Poly, dynamical systems and Moore machines as lenses, and polynomial comonoids as categories.
Sources: the paper, arXiv:2312.00990v2, checked against the arXiv listing. Index: Papers.
Key definitions and results
- Definition 2.1: polynomial functor
- Definition 2.8: positions and directions
- Definition 3.1: dependent lens; the category
- Definition 3.65: parallel product
- Definition 4.1: Moore machines as lenses