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

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

[Rory Lucyshyn-Wright]

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/



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