The category of Colombeau algebras

Since the beginning of Colombeau's the theory of algebras of generalized functions, the role of its characteristic polynomial growth versus a more general condition has been explored. Recently, we introduced the notion of asymptotic gauge (AG), and we used it to study Colombeau AG-algebras. This construction concurrently generalizes many different algebras used in Colombeau's theory and, at the same time, allows for more general growth scales. In this paper, we study the categorical properties of Colombeau AG-algebras with respect to the choice of the AG. The main aim of the paper is to study suitable functors to relate differential equations framed in algebras having different growth scales.