8.1 Constructed
- State-dependent family $\{\Pi_x\}_{x \in X}$ with $\Pi_x^2 = \Pi_x$.
- Disjoint union $\bigsqcup_x \{\Pi_x, \mathrm{id}_X\}$.
- Operational monoid $\mathcal{R}$ of witness-graph transitions.
8.2 Not Constructed
A monad on a category $\mathcal{C}$ is a triple $(T, \eta, \mu)$ where $T : \mathcal{C} \to \mathcal{C}$ is an endofunctor, $\eta : \mathrm{Id} \Rightarrow T$ is the unit, $\mu : T^2 \Rightarrow T$ is the multiplication, with monad laws $\mu \circ T\mu = \mu \circ \mu T$ and $\mu \circ T\eta = \mu \circ \eta T = \mathrm{id}_T$. The idempotent family $\{\Pi_x\}$ does not by itself constitute a monad.
A functor $\mathcal{F} : \mathbf{Cat} \to \mathbf{Dyn}$ requires $\mathbf{Cat}$ and $\mathbf{Dyn}$ to be explicitly defined with objects, morphisms, identity, composition. Neither category is currently defined.
8.3 Required Next Steps
- Define $\mathbf{Cat}$ with objects $(X, \Pi)$ and morphisms $f$ satisfying $f \circ \Pi_1 = \Pi_2 \circ f$.
- Define $\mathbf{Dyn}$ with objects $(X, F, S)$ (manifold, vector field, switching surface) and morphisms.
- Construct $\mathcal{F}$ on objects and morphisms.
- Verify $\mathcal{F}(\mathrm{id}) = \mathrm{id}$ and $\mathcal{F}(g \circ f) = \mathcal{F}(g) \circ \mathcal{F}(f)$.
- (If aiming at a monad) Construct $T, \eta, \mu$ on the chosen base category.