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Nathanael Arkor (Jan 04 2021 at 17:56):Is there any established alternative name for what Carboni and Walters call a "cartesian bicategory"? I find this terminology very misleading, and it clashes with existing conventions.
Nathanael Arkor (Jan 04 2021 at 18:24):"Bicategory of generalised relations" captures the original motivation (a "bicategory of relations" was defined by Carboni and Walters to be a "Cartesian bicategory" for which every object was discrete in an appropriate sense), though I don't believe the term has actually been used in the literature.
Todd Trimble (Jan 04 2021 at 20:14):Nathanael Arkor said:
"Bicategory of generalised relations" captures the original motivation (a "bicategory of relations" was defined by Carboni and Walters to be a "Cartesian bicategory" for which every object was discrete in an appropriate sense), though I don't believe the term has actually been used in the literature.
If "cartesian bicategory" is lousy, then "discrete cartesian bicategory" is lousier still. These designations really refer to the maps = left adjoints in the Rel-like bicategory B: these are the 1-cells in a bicategory Map(B) that, under the cartesian bicategory axioms, have 2-products and local products, hence the word "cartesian". If B is moreover locally posetal -- a running simplifying assumption in the original Carboni-Walters paper -- then the so-called "discreteness condition" (which is really a Frobenius condition that the canonical 2-cell is an isomorphism) implies that the bicategory Map(B) is locally discrete, hence the term "discrete". But this terminology is hopelessly married to the locally posetal context, whereas C & W from the get-go wanted a notion which would capture things like Span and Prof.
For example, the monoidal bicategory B consisting of groupoids and profunctors between them satisfies the Frobenius condition, but Map(B) is the bicategory of groupoids and functors between them: not locally discrete.
The term "bicategory of relations" for a (locally posetal) discrete cartesian bicategory appears not only in the nLab, but recently in this paper by Evan Patterson, to name one example; others are easily Googleable. I myself think it's a nicer name for the concept.
Nathanael Arkor (Jan 04 2021 at 20:21):@Todd Trimble: thanks for those clarifications. I agree that the use of "discrete" there is also misleading. What term would you use for the non-locally-posetal non-discrete version that appears in Cartesian bicategories II, or do you think it's reasonable even to use "bicategory of relations" for this case too?
Todd Trimble (Jan 04 2021 at 20:26):I'd be leery to use "bicategory of relations" outside the locally posetal context: too much potential for misunderstanding (alas). Unfortunately, I don't have any great suggestions, but "Frobenius bicategory" (bypassing "discrete" and "cartesian") might not be absolutely horrible. :-)