The Hopf algebra of Möbius intervals

Abstract

An unpublished result by the first author states that there exists a Hopf algebra H such that for any Moebius category C (in the sense of Leroux) there exists a canonical algebra morphism from the dual H* of H to the incidence algebra of C. Moreover, the Moebius inversion principle in incidence algebras follows from a `master´ inversion result in H*. The underlying module of H was originally defined as the free module on the set of iso classes of Moebius intervals, i.e. Moebius categories with initial and terminal objects. Here we consider a category of Moebius intervals and construct the Hopf algebra via the objective approach applied to a monoidal extensive category of combinatorial objects, with the values in appropriate rings being abstracted from combinatorial functors on the objects. The explicit consideration of a category of Moebius intervals leads also to two new characterizations of Moebius categories.