I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", page 344). The article reads:
We define an umbral calculus $(C, r)$ to consist of a coalgebra $C$ in $\mathcal{C}_{R}$ [the category of coalgebras over a commutative ring $R$] and an evaluation functional $r$ in $C^*$ (the dual $\mathrm{Hom}_{R}(C, R)$). For added convenience we insist that $C$ be supplemented, in the sense that it is equipped with a summand of scalars whose projection is a counit $\epsilon: C\to R$, and that $r$ acts on this summand as an identity.
I am completely lost beginning at "that $C$ be supplemented". A coalgebra is a linear space $C$ with coassociative comultiplication $\Delta: C \to C \otimes C$ and counit $\varepsilon: C \to R$ satisfying the opposite diagrams for associativity of multiplication and unit in an associative unital $R$-algebra. What is meant here by scalars? Are scalars elements of the ring $R$? What is the summand of scalars? Summand with respect to what operation, what objects are being summed? Is it an operation or an object? What does it mean for a "summand" to have a "projection" that is a map from $C$ to $R$? Projection by what map? This terminology is neither explained nor used neither before nor after, and I wasn't able to google what is a "supplemented coalgebra".
This question was cross-posted to MO.
