ASI Polynomial Operads Skill

"Pattern runs on matter: The free monad monad as a module over the cofree comonad comonad" — Libkind & Spivak (ACT 2024)

1. Polynomial Functors (Spivak)

Core Definition

A polynomial functor $p: \text{Set} \to \text{Set}$ is a sum of representables:

$$p \cong \sum_{i \in p(1)} y^{p[i]}$$

Where:

• $p(1)$ = set of positions (questions, observations)
• $p[i]$ = set of directions at position $i$ (answers, actions)

Morphisms (Dependent Lenses)

A lens $f: p \to q$ is a pair $(f_1, f^\sharp)$: