Combinatorial Models for Coalgebraic Structures

Abstract

We introduce a convenient category of combinatorial objects, known as cell-sets, on which we study the properties of the appropriate free abelian group functor. We obtain a versatile generalization of the notion of incidence coalgebra, giving rise to an abundance of coalgebras, Hopf algebras, and comodules, all of whose structure constants are positive integers with respect to certain preferred bases. Our category unifies and extends existing constructions in algebraic combinatorics, providing proper functorial descriptions; it is inspired in part by the notion of CW-complex, and is also geared to future applications in algebraic topology and the theory of formal group laws.