A sheaf of bicomodules over the incidence coalgebra

Abstract

To each partition π of a locally finite partially ordered set P into convex subsets is associated a C(P)-bicomodule S( π) where C( P) is the incidence coalgebra of P over some commutative ring. The complete lattice of all partitions of P into convex subsets is given a structure of Grothendieck topology in such a way that the association of S( π) to π yields a sheaf of C(P)-bicomodules. Some applications of parts of this machinery are given including a proof of Rota's Main Theorem.