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Stream: learning: questions

Topic: Cocomplete fibrations

fosco (Nov 11 2023 at 09:01):

An object $X$ of a 2-category $\mathcal K$ is said to be "cocomplete" with respect to a KZ-doctrine $S : \mathcal K\to\mathcal K$ if every $g : B\to X$ has a left extension $\bar g$ along the unit $B\to SB$ and such left extension preserves all left extensions $\hat f$ of 1-cells $f : A\to SB$.

There is a Yoneda structure on $Cat$, with a presheaf construction $P$ for which being $P$-cocomplete, or a "$P$-algebra", amounts to be cocomplete in the usual sense.

fosco (Nov 11 2023 at 09:04):

More in general, there is a Yoneda structure on the 2-category of pseudofunctors $\mathcal A^o\to Cat$ (so, on the 2-category of fibrations over $\mathcal A$) for every category $\mathcal A$, regarded as a bicategory with only identity 2-cells, with presheaf construction $\hat P$. This is the content of Street, "Conspectus of variable categories."

My question is, how do the two induced notions of $P$-cocompleteness and $\hat P$-cocompleteness relate. It's a very old story that I would like to understand in full now.

fosco (Nov 11 2023 at 09:04):

John Gray writes in his La Jolla paper "Fibred and cofibred categories" that the following conditions are equivalent.

(i) the total category $E$ of a fibration $p : E\to B$ is cocomplete as an object of $Cat$ (i.e. the domain functor $Fib/B\to Cat : p\mapsto E$ factors through $P$-algebras) and $p$ preserves colimits.
(ii) Each fiber $E_b$ at an object $b$ is cocomplete and each inclusion $E_b\hookrightarrow E$ preserves colimits (this is an equivalent condition for "every reindexing preserves colimits", which is how it's usually phrased in, e.g., Jacobs' book, and appears as Prop. 4.1 of Gray).

fosco (Nov 11 2023 at 09:05):

On the other hand, "having a cocomplete total category" looks like a stronger condition than being "internally cocomplete", i.e. being a $\hat P$-algebra, and this suspect is confirmed reading Streicher: "[being] cocomplete in the sense of fibered categories is weaker than cocomplete in the sense of ordinary category theory". The problem arises when I can find a definition of what is a "cocomplete fibration" in Jacobs: he seems to define cocompleteness as follows:

A fibration $p : E\to B$ is cocomplete (Definition 1.9.11) if every reindexing $u^* : E_y\to E_x$ has a left adjoint $\coprod_u$, and if condition (ii) of Gray holds (Definition 1.8.1). Which seems in fact stronger than Gray's definition, because there might be reindexings without a left adjoint, and (ii) implies cocompleteness of the total category.

So, I am confused.

fosco (Nov 11 2023 at 09:06):

(Each of these messages seems to be fine when I send it separate from the rest, but I couldn't make it a single post: is there a problem with LaTeX rendering?!)

fosco (Nov 11 2023 at 09:20):

Anyway, the question, in case it is unclear from the wall of text, is this: since free cocompletion in Cat and free cocompletion in Fib might differ, I want to understand how the two notions of cocompleteness relate, and if the one in Fib is \hat P-cocomplenetess or not.

Streicher says that being cocomplete in Fib is weaker than in Cat, but Jacobs calls a fibration "cocomplete" if its domain is cocomplete + stuff. This confuses me.

Mike Shulman (Nov 11 2023 at 09:55):

First, are you sure that's what Gray wrote? The theorem along those lines that I know is that the total category of a fibration is complete with $p$ preserving limits if and only if each fiber is complete and the transition functors $E_b \to E_{b'}$ preserve limits. But if you dualize that to colimits, then the fibration must also dualize to an opfibration, plus the second statement is about transition functors rather than inclusion functors.

fosco (Nov 11 2023 at 10:01):

image.png

This is Gray, apparently you're right on dualizing fibrations to opfibrations; but then, when is the total category of a fibration cocomplete (and when is the total category of an opfibration complete)?

4.1 says that having inclusions of each fiber continuous means having continuous reindexings.

fosco (Nov 11 2023 at 10:02):

(And I suspect that dualizes to cocontinuous?)

Mike Shulman (Nov 11 2023 at 10:03):

Second, I think the point is that there are two different perspectives on (co)completeness for a fibration. On the one hand, if you want to talk about limits indexed by ordinary categories, then you get a notion like the one in my last message for limits. For colimits, usually the appropriate version of this condition is the second one, that the fibers are cocomplete and the reindexing functors preserve colimits. I don't think it's that interesting to ask whether the total category of a fibration is cocomplete in its own right.

But on the other hand, it's also natural to instead view the base of the fibration as a replacement for the category of sets (e.g. this is what happens when doing "category theory over a base topos"), in which case one wants to consider limits indexed by internal categories in the base category. The special case of this for coproducts indexed by objects of the base category corresponds to the existence of left adjoints to reindexings, and from that you can construct the general case if you also have finite colimits in the external sense (fibers have finite colimits preserved by reindexing).

Mike Shulman (Nov 11 2023 at 10:04):

fosco said:

4.1 says that having inclusions of each fiber continuous means having continuous reindexings.

(And I suspect that dualizes to cocontinuous?)

Ah, yes. That'll dualize to cocontinuity, for opfibrations.

Mike Shulman (Nov 11 2023 at 10:06):

Finally, you can also combine the two, asking for left adjoints to reindexings (plus Beck-Chevalley) as well as external cocompleteness (for all ordinary small categories, not just finite ones). This is equivalent to having (weighted) limits indexed by all "multi-object internal categories" in the base category.

Mike Shulman (Nov 11 2023 at 10:07):

There's a discussion of this in a more general context in my paper Enriched indexed categories . I used proarrow equipments because I like that language better, but it should be transferable to Yoneda structures.

fosco (Nov 11 2023 at 10:30):

Mmmh, thanks, I will have to meditate on this. But then, I must deduce that there is no pre-packaged answer to "which notion of cocompleteness is captured by the Yoneda structure on fibrations"?

I might have to read your paper indeed...

fosco (Nov 11 2023 at 10:52):

I don't think it's that interesting to ask whether the total category of a fibration is cocomplete in its own right.

Why?

fosco (Nov 11 2023 at 11:02):

So, I read your message again and I agree that there are these two perspectives that one can take: but which notion of cocompleteness is captured by the presheaf construction on Fib? the unit of the presheaf construction on Cat, whiskered with the domain functor $d :Fib\to Cat$, gives a 2-cell $d\hat P\Rightarrow Pd$ which induces a functor from $P$-algebras to $\hat P$-algebras (provided it's a distributive law between monads, which I believe considering its canonical origin). This means that I can see a cocomplete category as a $\hat P$-cocomplete fibration, but not the other way round.

Mike Shulman (Nov 11 2023 at 17:40):

All three of the notions of cocompleteness that I mentioned have their own corresponding presheaf constructions. In fact the first two sit inside the third. This is also discussed in my paper. I don't know which of them is the one that Street constructed.

Mike Shulman (Nov 11 2023 at 17:41):

fosco said:

I don't think it's that interesting to ask whether the total category of a fibration is cocomplete in its own right.

Why?

Maybe I should say, I don't think it's that interesting to ask for that unless the fibration is also an opfibration.

fosco (Nov 11 2023 at 18:06):

All three of the notions of cocompleteness that I mentioned have their own corresponding presheaf constructions.

Interesting; but does it mean that there are three yoneda structures on fibrations?
I really have to read you, but sigh... who has time? :frown:

Street one is like this, iirc: take a fibration $p : E\to B$, take the presheaf category of each fiber $PE_b$, and glue the resulting pseudofunctor back into a fibration $\hat Pp : \hat E \to B$ over the same base.

Mike Shulman (Nov 11 2023 at 21:08):

I would guess all three of them are Yoneda structures, but I was never really motivated to understand the definition of Yoneda structure so I can't say for sure if they all satisfy the definition. Yoneda structures treat size differently than equipments, and I prefer the equipment approach, so it could be that something breaks in one or two of the cases when trying to reformat it as a Yoneda structure.

Mike Shulman (Nov 11 2023 at 21:08):

Your description of Street's construction sounds like it would probably correspond to the purely external/fiberwise notion of cocompleteness.

Mike Shulman (Nov 11 2023 at 21:10):

Although that'd be curious, since I seem to recall somewhere Street also commenting that an interesting example of a complete/cocomplete "variable category" is what we'd nowadays call a derivator, and those only have adjoints to reindexing and not fiberwise co/limits.

fosco (Nov 11 2023 at 21:24):

Mike Shulman said:

Although that'd be curious, since I seem to recall somewhere Street also commenting that an interesting example of a complete/cocomplete "variable category" is what we'd nowadays call a derivator, and those only have adjoints to reindexing and not fiberwise co/limits.

I know for a fact that the definition in "conspectus" is not very suitable to induce a yoneda structure on the 2-category of derivators. I tried, for very long time, to find one, and couldn't (I got this close, and then started doing something else: please, steal my few ideas and do it on my behalf ;-) ) https://www.youtube.com/watch?v=kLR0GIPWL7w

fosco (Nov 11 2023 at 21:28):

(The problem is precisely the one you point out.)

Back then I was stubborn, I wanted a yoneda structure at all costs, and ultimately I didn't know enough formal CT to tackle the problem from all angles. Probably there is a way to equip $\bf Der$ with proarrows...

Mike Shulman (Nov 11 2023 at 21:55):

The 2-category of derivators isn't going to be very well-behaved, since derivators include co/completeness conditions. They should instead be the co/complete objects inside a 2-category of prederivators.

Mike Shulman (Nov 11 2023 at 21:56):

And a prederivator is just a Cat-indexed CATegory, so there's certainly a presheaf Yoneda/equipment structure there. I expect the tricky part would be incorporating the 2-categorical structure of Cat correctly.

Christian Williams (Nov 11 2023 at 21:58):

My thesis defines relations of bifibered categories, and these form an equipment, fibered over the equipment of categories.

Just as a fibered category is a module of an arrow double category, a "fibered profunctor" is a module of an "arrow vertical profunctor".

But these do not compose. Instead, it is bifibered categories which have profunctors that compose. In the thesis they are called "matrix profunctors".

fosco (Nov 11 2023 at 22:01):

Mike Shulman said:

The 2-category of derivators isn't going to be very well-behaved, since derivators include co/completeness conditions. They should instead be the co/complete objects inside a 2-category of prederivators.

Of course, yes, I meant exactly this... It's been a long day ;-) it's night in Europe. A yoneda structure on PDer should recognize derivators as some special cocomplete objects (having homotopy co/limits is only one of the axioms). Only-recently published work by Marelli goes in this direction, recognizing Der as a sketchable 2-category in some specific sense. I wanted to account for various operational notions of locally presentable derivators (Rénaudin, Cisinski et al) building on this https://www.sciencedirect.com/science/article/abs/pii/S0022404922001517

I better go to bed now!

fosco (Nov 11 2023 at 22:06):

there's certainly a presheaf Yoneda/equipment structure there

Sure, the problem of that yoneda structure is that it has no homotopical meaning (it recognizes a notion of accessibility and cocompleteness that is not the homotopy-correct one). Let's go back at this tomorrow. I still have some questions. Thanks again!

fosco (Nov 13 2023 at 12:04):

The thing I still don't understand is the statement of what is called 4.1 above:
image.png

too bad that, if $cod : C^\to \to C$ is the codomain functor, a fibration if C has finite limits, including a fiber $C/b$ into $C^\to$ does not preserve the terminal object. What am I missing?