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Stream: theory: category theory

Topic: Reference in Cartesian Bicategories I

Robin Piedeleu (Nov 07 2021 at 18:17):

In Cartesian Bicategories I, Carboni & Walters state the following very nice result: "bicategories of relations of elementary toposes can be characterized as those functionally complete bicategories of relations such that $B(X, - )$ is representable in Map($B$)", where $B$ is a bicategory of relation and Map($B$) its associated category of maps (i.e. left adjoints).

They then add that "A good set of operations and equations to express this representability has been found by Freyd", but do not provide a reference. My guess is that this is a result in the world of allegories, but I am too unfamiliar with that world to figure it out.

I'd be really grateful if someone could help me find out what this mysterious "good set of operations and equations" is and where it is located.

Robin Piedeleu (Nov 07 2021 at 18:20):

I should probably add that they do cite a 1974 paper by Freyd, On canonizing category theory, or, On functorializing model theory, but I could not find it either!

Nathanael Arkor (Nov 07 2021 at 18:21):

This is not an answer, but in my experience, the phrase

has been found by [so-and-so]

without a citation often indicates the result never appeared in the literature. You could always send Freyd an email to enquire directly.

Robin Piedeleu (Nov 07 2021 at 18:22):

Yeah, I'm afraid this might be one of these cases.

Nathanael Arkor (Nov 07 2021 at 18:22):

Robin Piedeleu said:

I should probably add that they do cite a 1974 paper by Freyd, On canonizing category theory, or, On functorializing model theory. A Pamphlet, but I could not find it either!

Oh, I seem to recall that I also looked for this paper without success. But I didn't try emailing him directly.

Robin Piedeleu (Nov 07 2021 at 18:38):

Just emailed him - I'll let you know!

Mike Shulman (Nov 07 2021 at 18:41):

I would have assumed this was just a reference to the theory of [[allegories]].

Chad Nester (Nov 07 2021 at 19:38):

I think this is called a "power allegory" in Categories, Allegories.

Robin Piedeleu (Nov 07 2021 at 20:36):

Thanks Mike and Chad, power allegories seem to be what I'm looking for.

If anyone is interested, they are allegories for which the inclusion of maps $\text{Map}(B)\rightarrow B$ has a right adjoint. Since, apparently, every bicategory of relations is also an allegory, we recover the representability result quoted above.

Robin Piedeleu (Nov 07 2021 at 20:38):

Now I just need to find out about their axiomatisation from Categories, Allegories...

Nathanael Arkor (Nov 07 2021 at 20:43):

Robin Piedeleu said:

Thanks Mike and Chad, power allegories seem to be what I'm looking for.

If anyone is interested, they are allegories for which the inclusion of maps $\text{Map}(B)\rightarrow B$ has a right adjoint. Since, apparently, every bicategory of relations is also an allegory, we recover the representability result quoted above.

It seems like an interesting property to ask for. Has this notion been studied for general bicategories anywhere?

Mike Shulman (Nov 08 2021 at 00:28):

Personally, I prefer chapter A3 of Sketches of an Elephant as a place to read about allegories. The notation and terminology in Categories, Allegories is often nonstandard and hard to follow.

Mike Shulman (Nov 08 2021 at 00:30):

Nathanael Arkor said:

It seems like an interesting property to ask for. Has this notion been studied for general bicategories anywhere?

@Todd Trimble once wrote something about bicategories with a property like this, calling them epistemologies.

Todd Trimble (Nov 08 2021 at 00:38):

Indeed, when I was young, I was interested in whether there could be compact cartesian closed bicategories $B$ with this property, that the inclusion $i: \text{Map}(B) \to B$ could have a right biadjoint $P: B \to \text{Map(B)}$. If my memory is correct, Mike seemed to think this was possible. I regret not taking Mike up on this further, but I would love to resolve this question.

Nathanael Arkor (Nov 08 2021 at 00:41):

Thanks for pointing that out! From the page, V-Prof, for V a commutative quantale, is a motivating example. Presumably this breaks down somehow when V is a nice symmetric monoidal category more generally if that wasn't included as an example?

Mike Shulman (Nov 08 2021 at 00:44):

It's been a long time since I thought about this sort of thing.

Mike Shulman (Nov 08 2021 at 00:48):

In general the problem with non-posetal examples is size, of course. The right adjoint is the "presheaf category", but when $V$ is large then that goes up in size; while when $V$ isn't a poset, it's hard for it to be (co)complete without being large.

Mike Shulman (Nov 08 2021 at 00:48):

So the obvious place to look for an example would be enrichment over a [[small complete category]], like modest sets in a realizability topos.

Nathanael Arkor (Nov 08 2021 at 00:50):

Is it potentially the kind of size issue that could be addressed by solutions like small presheaves or relative adjoints, or does that turn out not to work, or be inelegant here?

Mike Shulman (Nov 08 2021 at 00:52):

I don't immediately see a way to use those tools, but I haven't thought about it for more than a few seconds, so I wouldn't rule it out.

Nathanael Arkor (Sep 20 2022 at 21:02):

Robin Piedeleu said:

Just emailed him - I'll let you know!

Did you ever get a response @Robin Piedeleu?

Robin Piedeleu (Sep 21 2022 at 09:06):

Unfortunately not!

Nathanael Arkor (Sep 21 2022 at 18:03):

Alas!