Linking diagrams for free

TL;DR

This paper constructs a category of linking diagrams using injective relations, unifying various diagrammatic structures like Temperley-Lieb, Brauer, and partition monoids, and showing they arise naturally from this framework.

Contribution

It introduces the category Link=Span(iRel), demonstrating that many linking and partition structures are instances within this category, providing a unified and free construction for these diagrams.

Findings

Link contains the free compact closed category on a self-dual object.

Includes Jones-Martin partition monoids within Link.

Path composition is achieved via pullback, accommodating loops and partiality.

Abstract

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.