Polynomial functors and polynomial monads — Nicola Gambino & Joachim Kock (2009; Math. Proc. Cambridge Philos. Soc. 154 (2013)). arXiv:0906.4931 (v2, PDF).
Develops polynomial functors over locally cartesian closed categories: a polynomial is a diagram , and its functor is pullback, then dependent product, then dependent sum. Polynomials compose, have strengths and preserve connected limits; natural transformations between them are represented by diagrams; they form a double category and a bicategory; and polynomial monads, including those generated by W-types, are studied.
Sources: the paper, arXiv:0906.4931v2, checked against the arXiv listing. Index: Papers.
Key definitions and results
- §1.4: polynomials and polynomial functors
- Proposition 1.12: composition
- Corollary 1.14: the smallest class closed under pullback, , and composition
- Propositions 1.15–1.16: strength; preservation of connected limits
- Proposition 1.22: characterisation over Set
- Theorem 1.24: skeleton of the category of finite polynomials = the Lawvere theory of commutative semirings
- Theorem 2.12: representation of strong natural transformations
- §4.3: W-types as initial algebras of polynomial endofunctors
Concept notes
Polynomial Functor, Lawvere Theory