Universal properties of spaces of generalized functions

TL;DR

This paper explores the universal properties of various spaces of generalized functions, demonstrating their fundamental role in solving mathematical problems and providing characterizations up to isomorphism.

Contribution

It introduces a unified, category-theoretic framework for understanding the universal properties of spaces like Schwartz distributions, Colombeau algebra, and generalized smooth functions.

Findings

Characterization of Schwartz distributions via co-universal property

Colombeau algebra as a quotient algebra with infinitesimal representatives

Generalized smooth functions as universal maps for non-Archimedean nets

Abstract

By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal property of Schwartz distributions, as the simplest way to have derivatives of continuous functions, Colombeau algebra as the simplest quotient algebra where representatives of zero are infinitesimal, and generalized smooth functions as the universal way to associate set-theoretical maps of non-Archimedean numbers defined by nets of smooth functions (e.g. regularizations of distributions) and having arbitrary derivatives. Each one of these properties yields a characterization up to isomorphisms of the corresponding space. The paper requires only the notions of category, functor, natural transformation and Schwartz's distributions, and introduces the notion of universal solution using a simple and non-abstract language.