Linking diagrams for free
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 (revers...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.0805.1441 |