Categorical description of umbral calculus?

Question

The theme of my current research project is related to umbral calculus in the context of more general algebraic structures, like Hopf algebras (and, in particular, shuffle algebras), so I am trying to read everything I can find on the connections between these fields. I remember distinctly reading, a substantial time ago, in some introductory article I can't find now, that Gian-Carlo Rota gave a categorical description of umbral calculus at some point. However, despite having read more than a dozen of articles and books on the subject, I couldn't find a single mention even of the concept of category. The closest thing I could find is the seemingly seminal 1979 article by Rota and Joni, where a brief description of umbral calculus is given in terms of Hopf algebras. Could anyone be so kind as to indicate an article or book where umbral calculus is described from the categorical point of view, that is, if such a description exists?

Answer 1

An overview of the categorical description of umbral calculus is given by Nigel Ray, in Universal Constructions in Umbral Calculus (1996).

The most flexible basis for the study of umbral calculus lies in the category ${\cal C}_R$ of coassociative coalgebras over $R$, together with the category ${\cal A}_R$ of dual algebras. Additional features such as gradings and Hopf algebra structures may naturally be present in certain circumstances. This viewpoint was conceived by Joni and Rota (1979) and developed by Nichols and Sweedler (1982).