2008/05/11 by Dominic J. D. Hughes, Hughes, Dominic J. D.
Computer Science · Mathematics · #16B50 #18B10 #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #Mathematical Physics (math-ph) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0805.1441
openalex publication_date 2008/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Linking diagrams with path composition are ubiquitous, for example: Temperley-Lieb and Brauer monoids, Kelly-Laplaza graphs for compact closed categories, and Girard's multiplicative proof nets. We construct the category Link=Span(iRel), where iRel is the category of injective relations (reversed partial functions) and show that the aforementioned linkings, as well as Jones-Martin partition monoids, reside inside Link. Path composition, including collection of loops, is by pullback. Link contains the free compact closed category on a self-dual object (hence also the looped Brauer and Temperly-Lieb monoids), and generalises partition monoids with partiality (vertices in no partition) and empty- and infinite partitions. Thus we obtain conventional linking/partition diagrams and their composition "for free", from iRel.