Dear Colleagues, My PhD thesis is now available, as follows. Title: "Riesz-Schwartz extensive quantities and vector-valued integration in closed categories" URL: http://arxiv.org/abs/1307.8088 Defended June 18, 2013, York University, Toronto, Canada Supervisor: Walter Tholen External examiner: F. William Lawvere Abstract: "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 spaces, and we study R-module objects therein (or algebras of a commutative monad), which we call linear spaces. Building on ideas of Lawvere and Kock, we study functionals on the space of scalar-valued maps, including compactly-supported Radon measures and Schwartz distributions. We develop an abstract theory of vector-valued integration with respect to these scalar functionals and their relatives. 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 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 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." This work was presented at the following conferences: Samuel Eilenberg Centenary Conference, Warsaw 2013 21st Workshop on Foundational Methods in Computer Science, Sackville, 2013 Canadian Annual Symposium on Operator Algebras and their Applications, Toronto 2013 Your comments and questions are welcome. Regards, Rory Lucyshyn-Wright http://www.math.yorku.ca/~rorylw/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]