  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 distinct objects c,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 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 updown categories of integer partitions, integer compositions, planar
rooted trees, and rooted trees.
