Abstract:
We study compactly-supported Radon measures, Schwartz distributions, and their generalizations in an abstract setting through monoidal and enriched category theory, developing a theory of vector-valued integration with respect to these scalar functionals. Building on ideas of Lawvere and Kock, we work in a given closed category, whose objects we call spaces, and we study R-module objects therein (or algebras of a commutative monad), which we call linear spaces. For each space, we consider the space of scalar functionals on the associated space of scalar-valued maps. For appropriate choices of category, the latter functionals specialize to the classical notions of compactly-supported Radon measure and Schwartz distribution. We study three axiomatic approaches to vector integration, including an abstract Pettis-type integral, showing that all are encompassed by an axiomatization via monads and that all coincide in suitable contexts. We study the relation of this vector integration to novel relative notions of completeness in linear spaces. One such notion of completeness, defined via enriched orthogonality, characterizes exactly those separated linear spaces that support the vector integral. We prove Fubini-type theorems for the vector integral. Further, we develop aspects of several supporting topics in category theory, including enriched factorization systems, enriched associated idempotent monads and adjoint factorization, symmetric monoidal adjunctions and commutative monads, and enriched commutative algebraic theories