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 $\mathbb P^1_C\smallsetminus\{0,\infty\}$, where $C$ is algebraically closed and of characteristic zero. It aims at reading the existing classification results as an equivalence between regular-singular systems and representations of the group $\mathbb Z$. In the second part, we deal with regular-singular connections over $R((x))$ and $\mathbb P_R^1\smallsetminus\{0,\infty\}$, where $R=C[[t_1,\ldots,t_r]]/I$. The picture we offer shows that regular-singular connections are equivalent to representations of $\mathbb Z$, now over $R$.