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After a shamefully long time, I am working some more on cartesian bicategory; I have added some material on the locally cartesian structure, on the essential uniqueness of a cartesian structure on a bicategory, and a beginning of a section on the “Frobenius conditions”.
I also inserted a little promissory note acknowledging that it really would be better to deal with framed cartesian bicategories, by tweaking the definition a little. It would require a certain amount of rewriting (which makes me believe that I had better do it sooner than later).
A few days ago here at the nForum, I outlined a context where these Frobenius conditions imply “Frobenius reciprocity” (in response to a query of David Corfield). I want to see whether I can write out or at least sketch a proof in the context of a cartesian bicategory satisfying the Frobenius conditions, and see what else might be said on the relationship between the two Frobenii.
it really would be better to deal with framed cartesian bicategories, by tweaking the definition a little. It would require a certain amount of rewriting (which makes me believe that I had better do it sooner than later).
I hope you don’t get round to it before I do. I’ve also been trying to get at the right notion of a ’cartesian equipment’ (some very sketchy notes are on my personal web). Part of the problem is that there are at least three definitions of ’equipment’ (four if you count double categories with connections), and two notions of adjunctions between them (namely Grandis–Paré’s, for double categories, and Carboni–Kelly–Verity–Wood’s, for their notion of equipment). So what I’ve been (very slowly) doing is trying to explain why all those definitions of equipments are equivalent, with the aim of then showing that the two notions of adjunction coincide and then defining a cartesian equipment to be a cartesian object in a suitable (- or) tricategory of equipments. Equipments should be certain pseudomonads (on locally discrete bicategories) in the tricategory of biprofunctors.
But with the right notion of cartesian equipment, your stuff already at cartesian bicategory should then give an easy proof that is a cartesian bicategory if and only if is a cartesian equipment.
Finn, please be my guest! In fact, I wasn’t looking forward to doing this, because I am hardly expert in these matters. You are clearly much better qualified.
I hope it won’t be too much of a pain to do this; the tack I’ve so far taken at cartesian bicategory might be seen as somewhat idiosyncratic. I may continue just developing stuff over there for a while without trying to rewrite this in terms of equipments or whatever.
Well, I certainly didn’t mean to ask you not to work on it – I hope it didn’t come across that way. It’d be great if we could work together on this, or at least avoid duplicating each other’s efforts. I’ll keep working away at it on my personal web (this’ll be going into my thesis, so I want to keep a clear record of attribution), and if I come up with anything interesting I’ll post here. I’d be delighted if you had any thoughts to share on the subject.
No, it didn’t come across that way; furthermore, I’m very glad you’re interested! It’s something that we (and, I believe, Mike) agree ought to be done.
The fact that you’ve thought about this comes as a relief. As I implied, I don’t know the literature on this, and my own approach would have been to write down something naive and hope an expert could look it over. The nLab page on equipments looks straightforward enough and adaptable to cartesian structure. Maybe what I could do is try writing down here such a naive definition, and if you think it looks alright, it could be ported over to the page on a trial basis, as it were. (But I don’t want this to be a distraction from the more pressing needs you may have regarding thesis work.)
That sounds good. Following Cartesian bicategories II (prop. 4.2), my naive definition would be that an equipment, given as a span , is cartesian if and have finite products, and the adjunctions commute with the projections from to . But that’s only an educated guess – I haven’t really thought that far ahead.
Yes, someone should do this! And I’m perfectly happy if the someone isn’t me; I have plenty on my plate at the moment. But I’ll be watching interestedly. My guess is that a cartesian equipment should be just a cartesian object in the 2- or 3-category of equipments. I think that this is the approach taken in the papers “A 2-categorical approach to change of base and geometric morphisms” I (by Carboni, Kelly, and Wood) and II (by Carboni, Kelly, Verity, and Wood), although their notation is a bit hard for me to follow, and their notion of “equipment” not quite the same. They do mention the connection to cartesian bicategories, however.
Thanks, Mike! That sounds like a perfectly reasonable approach, but I’d have to think of exactly how it would go. It might require more reorganization than I really feel like undertaking at the moment. Without having looked properly at the literature, my understanding is that that fits the approach taken by some of those authors toward defining cartesian bicategories concretely in terms of 2-products and local cartesian products, as touched upon in the most recent additions to cartesian bicategory.
(That’s not to dismiss what Finn suggested – it’s just that I wasn’t quite following the suggestion, which is surely my fault. I probably just need to do some reading.)
Of course, one would hope that the two purported definitions would turn out to be equivalent. In what I said above, is the category of vertical arrows of a framed bicategory and is the category of horizontal arrows and 2-cells. Then finite products on the former, via adjunctions , should correspond to the products in the bicategory of maps of a cartesian bicategory, products on should be the local products, and having the source and target maps commute with these should correspond to the extra conditions on how the products and local products in a cartesian bicategory interact. That’s a lot of ’should’s, though, and I’m not sure if there’d be issues when is not locally discrete.
Thanks very much for that explanation, Finn – that helped enormously.
In section 8 of my paper on framed bicategories, there is a discussion of adjunctions thereof, and a rephrasing in terms of universal arrows and 2-cells. I don’t think I wrote this down, but that obviously implies a “local” characterization of what it means for a framed bicategory to be cartesian.
Interesting – I’d forgotten about that. Looking at it again, it seems to be much the same as (the dual of) theorem 3.6 in Grandis & Paré’s Adjoints for double categories.
Yes, it is.
I watched Todd’s talk at the MIT categories seminar, and I asked about the relationship between cartesian bicategories and cartesian proarrow equipments. He suggested I continue the discussion here. It looks like a discussion was started here, but never was resolved. I have to say I’m surprised this kind of thing has never been written up since it seems that Wood worked on both Cartesian Bicategories and Proarrow Equipments so you’d hope the relationship would be well understood.
Here’s my hope for what the relationship is. First, note that from every Bicategory we can make a Proarrow Equipment whose horizontal morphisms are taken from the bicategory and the vertical morphisms are adjoint pairs (maybe this only works with a strict 2-category though if we want vertical morphisms to be strict?). Let’s call this construction Adj, it’s probably a functor and it’s probably right adjoint to the forgetful functor Hor from proarrow equipments to bicategories.
I suspect/hope that a Cartesian bicategory is precisely a bicategory C such that Adj(C) is a cartesian proarrow equipment, i.e. a cartesian object in the 2-category of proarrow equipments, lax functors and vertical transformations.
Even better would be if this extends to the 2-category of cartesian bicategories, which would say that the 2-category of Cartesian Bicategories is the pullback of Adj : CartBicat -> ProarrowEquipment and U : CartProarrowEquipment -> ProarrowEquipment.
This would suggest that Cartesian Proarrow Equipments are the more natural object of study, since they capture some nice examples like Categories/Functors/Profunctors that aren’t captured by the other definitions. There are also probably reasons to study cartesian virtual equipments instead as well, but that’s a little off topic here.
That conjecture has been my expectation for a while as well.
Oh I see now looking back this is essentially what Finn was saying in #2.
sarahzrf points out on zulip that the page cartesian bicategory lists Prof as both an example and a non-example. I assume this is a mistake, and that it should be an example (except in the separately-listed case of non-cartesian enrichment), right?
Unfortunately, I can’t edit the page to fix this; I get the error “Invalid LaTeX block” without any indication of where the offending block is.
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