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Morgan Rogers (he/him) (Oct 25 2023 at 09:07):The fact that one can construct a category from a profunctor in several ways is interesting, but does anyone know of people working in the collage of a profunctor that I might be able to cite? In particular, there is a certain type of diagram that comes up when one searches for "collage of a profunctor", and I would like to know if people have used such diagrams for reasons other than to illustrate how the collage is constructed.
Jade Master (Oct 25 2023 at 10:29):I am curious what diagram you mean. For rererences, I only know of the nlab page https://ncatlab.org/nlab/show/cograph+of+a+profunctor
Matteo Capucci (he/him) (Oct 25 2023 at 13:52):I might say something completely bonkers, but isn't [[Artin gluing]] a special instance of collage?
Morgan Rogers (he/him) (Oct 25 2023 at 14:33):Jade Master said:
I am curious what diagram you mean. For rererences, I only know of the nlab page https://ncatlab.org/nlab/show/cograph+of+a+profunctor
Ha that was confusing, you put the full stop in the link and the nlab was baffled
Bryce Clarke (Oct 25 2023 at 14:36):Matteo Capucci (he/him) said:
I might say something completely bonkers, but isn't [[Artin gluing]] a special instance of collage?
I recall something like this is considered in the paper The glueing construction and double categories by Susan Niefield.
Morgan Rogers (he/him) (Oct 25 2023 at 14:43):Jade Master said:
I am curious what diagram you mean. For rererences, I only know of the nlab page https://ncatlab.org/nlab/show/cograph+of+a+profunctor
When you "draw a profunctor", as Fong and Spivak do in Seven Sketches you can interpret the result as a category, and this category is precisely the collage of the profunctor. For larger categories, one could imagine using a similar diagram to illustrate a diagram in the collage; this is the sort of diagram I'm talking about.
Dylan Braithwaite (Oct 25 2023 at 19:59):In section 2 of this paper by @Mario Román and I, we discuss a special case of a categorified version of this. Namely that string diagrams for biactegories/Cat-bimodules can be viewed as diagrams for the collage of a 2-profunctor.
The rest of the paper is loosely based on the idea that various extensions to monoidal string diagrams can be considered as diagrams for a lax colimit of diagrams of 2-profunctors, but we didn't get to tell that story as coherently as I would have liked due to time and space constraints.
Morgan Rogers (he/him) (Oct 26 2023 at 12:00):Thanks Dylan, that's interesting, especially the way that the boundaries of regions representing different categories can be flexible. While I will often be dealing with categories having products or coproducts, there isn't a canonical choice of monoidal product to hand that would justify a string diagrammatic presentation but this is a good reference for at least one other way that people present data in a collage :)