The category of Colombeau algebras

TL;DR

This paper explores the categorical properties of Colombeau AG-algebras, a unified framework that generalizes many existing Colombeau algebras, and examines their implications for solving generalized ODEs.

Contribution

It analyzes the categorical aspects of Colombeau AG-algebras and demonstrates their impact on the solvability of generalized ordinary differential equations.

Findings

Colombeau AG-algebras unify various Colombeau constructions.

Categorical properties influence the solvability of generalized ODEs.

The framework generalizes multiple existing Colombeau algebras.

Abstract

In [11], we introduced the notion of asymptotic gauge (AG), and we used it to construct Colombeau AG-algebras. This construction concurrently generalizes that of many different algebras used in Colombeau's theory, e.g. the special one $\mathcal{G}^{\srm}$, the full one $\gse$, the NSA based algebra of asymptotic functions $\hat{\mathcal{G}}$, and the diffeomorphism invariant algebras $\gsd$, $\mathcal{G}^{2}$ and $\hat{\mathcal{G}}$. In this paper we study the categorical properties of the construction of Colombeau AG-algebras with respect to the choice of the AG, and we show their consequences regarding the solvability of generalized ODE.