New York Journal of Mathematics
Volume 18 (2012) 275-289


Will Grilliette

Scaled-free objects

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Published: April 23, 2012
Keywords: Banach space, Banach algebra, adjoint functor, free construction
Subject: 46M99, 46B99, 46H99

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.

Author information

Division of Mathematics, Alfred University, 109B Myers Hall, Alfred, NY 14802