Scaled-Free Objects

Details Scaled-Free Objects Scaled-Free Objects

2010-11-02

arxiv.org/abs/1011.0717v3

New York J. Math. 18 275-289 (2012)

Mathematics

Functional Analysis

15 pages

Scientific paper

In this work, I address a primary issue with adapting categorical and algebraic concepts to functional analytic settings, the lack of free objects. Using a "normed set" and associated categories, I describe constructions of normed objects, which build from a set to a vector space to an algebra, and thus parallel the natural progression found in algebraic settings. Each of these is characterized as a left adjoint functor to a natural forgetful functor. Further, the universal property in each case yields a "scaled-free" mapping property, which extends previous notions of "free" normed objects. In subsequent papers, this scaled-free property, coupled with the associated functorial results, will give rise to a presentation theory for Banach algebras and other such objects, which inherits many properties and constructions from its algebraic counterpart.

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