Sheaf coalgebras and duality. (Q2381636)

Let \(R\) be a commutative ring with identity. The functor from the category of sets to the category of \(R\)-coalgebras, sending a set \(X\) to the free \(R\)-module with basis \(X\), and the coalgebra structure given by making the elements of \(X\) grouplike, has a right adjoint, assigning to any \(R\)-coalgebra \(C\) the set of grouplike elements \(G(C)\). In this paper this result is extended to sheaves of sets on a smooth manifold \(M\), where \(R\) is replaced by the ring of compactly supported functions on \(M\). A duality between sheaves on \(M\) and a class of coalgebras over this ring is established. The results of this paper are then used [in \textit{J. Mrčun}, J. Pure Appl. Algebra 210, No. 1, 267-282 (2007; Zbl 1115.22003)], where they are extended to étale Lie groupoids and Hopf algebroids.