10.1 Universal decision learning

Let \(J:\mathcal{O}\to \mathcal C\) embed observed contexts into a larger context category, and let \(F:\mathcal{O}\to \mathcal D\) record local decision information. UDL uses a left Kan extension to generate or aggregate global candidates and a right Kan construction to enforce compatibility:

Kan extensions are determined by universal properties rather than a chosen coordinate representation [ Kan , 1958 , Mac Lane , 1971 , Riehl , 2016 ] . Pointwise left extensions take the familiar form

The target category and its enrichment determine what aggregation actually means.