2004/02/27 by Michael E. Hoffman, Hoffman, Michael E.
Computer Science · Mathematics · #05A17 #05C05 (Secondary) #06A07 (Primary) #18B35 #Advanced Algebra and Logic #Algebraic structures and combinatorial models #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics #Rings, Modules, and Algebras #math.CO #math.CT #msc:05A17 #msc:05C05 #msc:06A07 #msc:18B35
paper · pdf · doi:10.48550/arxiv.math/0402450
21 pages
arxiv created 2004/02/27 · openalex publication_date 2004/02/27 · arxiv updated 2009/12/01 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
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 not equal to c'. If we keep in place the latter axiom but allow for more than one morphism between objects, we can 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, which incorporates earlier notions of differential posets and weighted-relation posets. We also give a detailed account of ten examples, including the updown categories of integer partitions, integer compositions, planar rooted trees, and rooted trees.