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

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 (section XI).

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).