Algebraic theory of formal regular-singular connections with parameters

This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over C((x)) and PC1 \ {0, oo}, where C is algebraically closed and of characteristic zero. It aims to read the existing classification results as an equivalence between regular-singular systems and representations of the group Z. In the second part, we deal with regular-singular connections over R((x)) and PR1 \ {0, oo}, where R = C OE OE t1, ... , tr center dot center dot=I. The picture we offer shows that regular-singular connections are equivalent to representations of Z, now over R.