7.8 Decision modalities

The abstraction is useful because the same typed questions recur in decision problems that otherwise use different algorithms.

Planning.

The left Kan stage aggregates partial paths, costs, or reachable outcomes. The right stage enforces compatibility with goals and continuation values. An infinitesimal decision records how changes in costs, dynamics, or a start context alter both the candidate path family and the globally coherent plan. A shortest-path switch is a nonsmooth change of active cocone.

Constrained optimization.

Candidate generation supplies feasible proposals; consistency enforces constraints and optimality conditions. Tangent and normal cones distinguish directions along the feasible set from directions that violate it. Failure of regularity at an active-set change appears as a typed obstruction rather than an unexplained optimizer instability.

Games.

Local strategies and payoff responses generate candidate profiles, while best-response compatibility selects equilibria. An equilibrium derivative describes the response to a payoff or information perturbation only while the equilibrium branch remains regular. Bifurcation or multiplicity requires a set-valued tangent semantics and an admission rule that states how a branch is selected.

Online decisions.

History-indexed local losses are accumulated forward, while regret or feasibility comparators impose global tests. The tangent layer distinguishes sensitivity to the latest evidence from sensitivity to the comparator class itself. Drift can therefore be localized to the data stream, decision model, or admission contract.

Causal decisions.

A mechanism-changing probe can transport a decision across an interventional context only when an intervention semantics has been supplied. The Kan construction organizes extension; it does not certify identification. Infinitesimal causality can provide grounded local mechanism directions, while infinitesimal decisions studies how those directions change a downstream choice.