In basic combinatorics course we learn about the incidence algebra of a poset. Now i've read that one could start by defining the co-incidence algebra and then it is a fact that the dual vector space of a co-algebra has a canonical algebra-structure. In this case we arrive at the incidence algebra. My main question is then, is it more natural to start with the coalgebra? What additional insights do we gain by considering this dual structure?
Some further questions along the line: What would be more concrete motivations to study the coalgebra structure, lets say in combinatorics (regarding posets)? For example, does it give more insight into Möbius inversion? (which is something I only know in terms of the algebra).
Moreover, incidence algebras and group algebras are fundamental examples of Hopfalgebras, that is, they have an additional map, the antipode. Now I have read the antipode takes the role of Möbius inversion. I understand that in the way that the antipode for the incidence bialgebra is defined in a way such that it generalizes somehow the Möbius function. How is it defined and how to make this generalization exact? I have the same question for the group algebra. Reading the definition of the antipode $S(g) = g^{-1}$ without context does not tell me anything. What is the meaning (context) of this map? What is the point of the bi-algebra structure of the group algebra/resp. of the incidence algebra? In the end, there is the issue of duality. The common thing which generalizes both, group algebra and incidence algebra, is a category algebra (https://en.wikipedia.org/wiki/Category_algebra). Referring to the wiki-article, in what sense is the incidence-algebra style definition dual to the group-algebra style definition (the only difference I spot is the finiteness condition)? Does I understand it correctly, that for an infinite (discrete) group, the group algebra does not have a coalgebra structure?