Updown categories: Generating functions and universal covers

TL;DR

This paper introduces 'updown categories', a generalization of posets allowing multiple morphisms and automorphisms, and develops their theoretical framework with detailed examples including partitions and trees.

Contribution

It provides a formal definition and foundational theory for updown categories, extending poset concepts to include multiplicities and symmetries.

Findings

Defined updown categories and their properties

Developed a comprehensive theoretical framework

Presented ten detailed examples including partitions and trees

Abstract

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.