Updown categories: Generating functions and universal covers

A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c, c′) and Hom(c′, c) is nonempty for c 6= c′. If we keep in place the latter axiom but allow for more than one morphism between objects, we have a sort of generalized poset in which there are multiplicities attached to the covering relations, and possibly nontrivial automorphism groups. We call such a category an “updown category.” In this paper we give a precise definition of such categories and develop a theory for them. We also give a detailed account of ten examples, including the updown categories of integer partitions, integer compositions, planar rooted trees, and rooted trees.