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 the sets Hom(c,c′)Hom(c,c′) and Hom(c′,c)Hom(c′,c) is nonempty for distinct objects cc, c′c′. Retaining the latter axiom but allowing for more than one morphism between objects gives a sort of generalized poset in which there are multiplicities attached to the covering relations, and possibly nontrivial automorphism groups of objects. An updown category is such a category with an appropriate grading on objects. In this paper we give a precise definition of updown categories and develop a theory for them, including two types of associated generating functions and a notion of universal covers. We give a detailed account of ten examples, including updown categories of sets, graphs, necklaces, integer partitions, integer compositions, planar rooted trees, and rooted trees.