University of Birmingham > Talks@bham > Lab Lunch > String diagrams for cartesian bicategories and their Karoubi envelopes - and entailment systems

String diagrams for cartesian bicategories and their Karoubi envelopes - and entailment systems

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If you have a question about this talk, please contact Todd Waugh Ambridge.

Joint work with Drew Moshier

I’ll describe how string diagrams can present the category structure in a number of old results.

[1] – continuous dcpos can be got as rounded ideal completions of idempotent relations, objects of the Karoubi envelope of Rel. The corresponding morphisms between the domains are certain “non-deterministic maps”, the Kleisli maps for the lower power domain.

[2] – amongst cartesian bicategories, Rel appears as a bicategory of relations, having extra structure and properties that make it compact closed (in fact dagger closed).

[3] – the category of stably compact spaces and closed relations [4] is the Karoubi envelope of a category Ent. Its “cut” composition, which involves a redistributivity deriving from the sequent calculus, makes it hard to work with, but there were signs of a useful diagrammatic calculus.

I shall describe how compact closed cartesian bicategory structure is inherited by Karoubi envelopes and is amenable to reasoning with string diagrams. Ent is not a bicategory of relations. However, it has a different kind of structure, as “bicategory of entailments”, that makes it dagger closed.

[1] Vickers “Information systems for continuous posets” [2] Carboni and Walters “Cartesian bicategories I” [3] Vickers “Entailment systems for stably locally compact locales” [4] Jung, Kegelmann, Moshier “Stably compact spaces and closed relations“

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