Scaled-Free Objects

TL;DR

This paper introduces a new framework for constructing free normed objects in functional analysis, extending algebraic free object concepts to Banach algebras through scaled-free properties and adjoint functors.

Contribution

It develops a categorical approach to define scaled-free normed objects, enabling algebraic-like constructions in functional analytic settings.

Findings

Defined scaled-free normed objects via adjoint functors

Established universal properties for these scaled-free objects

Lays groundwork for presentation theory of Banach algebras

Abstract

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.