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 systemsAdd to your list(s) Download to your calendar using vCal - Dr Steve Vickers (School of Computer Science, University of Birmingham)
- Thursday 21 November 2019, 12:00-13:00
- CS 217.
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“ This talk is part of the Lab Lunch series. ## This talk is included in these lists:- CS 217
- Computer Science Departmental Series
- Computer Science Distinguished Seminars
- Lab Lunch
- Theoretical computer science seminar
- computer sience
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