C.29:13.4 - Applied category theory

Adopt the operation-preservation discipline explained by Fong and Spivak, §3.3.2: a functor maps objects and arrows while preserving identities and composition. C.29.1 makes the corresponding comparison available before a result is reused, including operation permissions, representative independence and weaker bounds. The authors’ §2.5.3 also shows how composition and choice compute route costs; C.29.1:5.2 uses an authored case to expose when an omitted continuation condition defeats that reduction.

Applied category theory remains one organizer for composition, interfaces and transfer. The book’s databases, electric circuits and dynamical systems provide source applications; adjoint functors, enriched categories and toposes provide further constructions to study when the question needs them. The trade-off is the work of defining the objects and operations and establishing their laws. When an ordinary domain calculation already provides the correspondence and consequence, use it directly. A failed comparison can instead identify the next required distinction or construction.