lin-0094

7.1 Decision making as universal extension

Let \(\mathcal O\) be a category of observed or locally accessible contexts, \(\mathcal C\) a larger category of contexts in which decisions must be made, and

\[ J:\mathcal O\longrightarrow \mathcal C \]

the functor that relates the two. A local decision model is a functor

\[ F:\mathcal O\longrightarrow \mathcal E, \]

where \(\mathcal E\) contains the relevant decision objects: values, distributions, feasible sets, policies, strategies, or enriched utilities.

The pointwise left Kan extension

\[ L_F=\operatorname {Lan}_J F \]

aggregates the locally available ways of reaching, explaining, or generating a context:

\[ L_F(c) \cong \operatorname *{colim}_{(J o\to c)\in (J\downarrow c)}F(o). \]

It is the candidate-generating side of decision making: rollout, interpolation, search, accumulation, or forward propagation.

The right Kan extension

\[ R_F=\operatorname {Ran}_J(J^\ast L_F), \qquad J^\ast L_F=L_F\circ J, \]

assembles what remains compatible with the restrictions, continuations, or tests visible from a context:

\[ R_F(c) \cong \operatorname *{lim}_{(c\to J o)\in (c\downarrow J)}L_F(Jo). \]

It is the coherence side: feasibility, backward consistency, equilibrium, or fixed-point semantics. In the simplest UDL presentation,

\[ \operatorname {UDL}_J(F) = \operatorname {Ran}_J \bigl(J^\ast \operatorname {Lan}_JF\bigr). \]

The universal comparisons are especially transparent in the graphical language for 2-categories introduced in Chapter 0. For the right-hand panel below, write \(G=J^\ast L_F\). The left Kan unit changes the direct local wire \(F\) into the composite \((\operatorname {Lan}_JF)J\); the right Kan counit changes the composite \((\operatorname {Ran}_JG)J\) back into \(G\). The two cells face opposite directions, which is the precise graphical content of their left–right duality.

The unit and counit comparison cells for left and right Kan extension. The intermediate context category appears as a region created or removed at the comparison cell.
Figure 7.1 The unit and counit comparison cells for left and right Kan extension. The intermediate context category appears as a region created or removed at the comparison cell.
Diagram illustrating 7.1 Decision making as universal extension.

This formula is a semantic pattern, not a claim that every decision algorithm uses the same numerical implementation. The indexing functors, enrichment, order of operations, and approximation scheme are part of the declaration. The universal properties specify what a correct extension must mediate; they do not select data, establish causal meaning, or guarantee that the required limits and colimits exist.

Design principle

A decision learner must expose both halves of its semantics: how candidates are generated and how global coherence is imposed. A final action can conceal failure in either half.