Riesz-Schwartz extensive quantities and vector-valued integration in closed categories

Rory Lucyshyn-Wright
Monday, July 22, 2013 - from 17:30 to 17:55
We develop aspects of functional analysis in an abstract axiomatic setting, through monoidal and enriched category theory. We work in a given closed category, whose objects we call \textit{spaces}, and we study \mbox{$R$-module} objects therein (or algebras of a commutative monad), which we call \textit{linear spaces}. Building on ideas of Lawvere and Kock, we study functionals on the space of \mbox{scalar-valued} maps, including \mbox{compactly-supported} Radon measures and Schwartz distributions. We develop an abstract theory of \mbox{vector-valued} integration with respect to these scalar functionals and their relatives. We study three axiomatic approaches to vector integration, including an abstract \mbox{Pettis-type} integral, showing that all are encompassed by an axiomatization via Eilenberg-Moore algebras and that all coincide in suitable contexts. We study the relation of this vector integration to relative notions of completeness in linear spaces. One such notion of completeness, defined via enriched orthogonality, determines a symmetric monoidal closed reflective subcategory consisting of exactly those separated linear spaces that support the vector integral. We prove \mbox{Fubini-type} theorems for the vector integral. Further, we develop aspects of several supporting topics in category theory, including enriched orthogonality and factorization systems, enriched associated idempotent monads and adjoint factorization, symmetric monoidal adjunctions and commutative monads, and enriched commutative algebraic theories.