The category of Colombeau algebras

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 ${\mathcal{G}}^{\text{srm}}$, the full one $\text{gse}$, the NSA based algebra of asymptotic functions $\widehat{\mathcal{G}}$, and the diffeomorphism invariant algebras $\text{gsd}$, ${\mathcal{G}}^2$ and $\widehat{\mathcal{G}}$. 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.