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 Colombeaus 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.
We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general growth condition formalized
by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restriction on the coefficients can be solved.
