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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?

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    $\begingroup$ You might try searching for "umbral calculus" and "species." I tried this just now and got some potentially relevant hits, such as the nForum and Möbius polynomial species by Senato et al. $\endgroup$ Commented Jan 4 at 18:10

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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).

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    $\begingroup$ Thank you! I will leave this question open for a while, in case if someone else decides to add more information, then accept the answer. $\endgroup$ Commented Jan 4 at 14:21
  • $\begingroup$ P.S. I've skimmed the article (have finally found this collection of essays), and it appears to be less of a category-theoretic treatment of the umbral calculus, more of a simple indication of the category in which the action takes place. Thank you nonetheless, not exactly what I'm looking for, but may be useful. $\endgroup$ Commented Jan 4 at 17:55

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