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Stream: theory: category theory

Topic: Constructing compact closed bicategories

Tobias Fritz (Nov 13 2023 at 08:44):

There are a few methods for constructing symmetric monoidal bicategories which bypass lengthy checking of coherences, in particular @Mike Shulman's constructions based on monoidal double categories. Unfortunately those results don't quite apply in my situation. However, I expect there to be a general recipe for building compact closed bicategories, which should apply in my case, based roughly on the following kind of data:

• An involutive symmetric monoidal category $\mathcal{C}$. I will write $\bar{\cdot} : \mathcal{C} \to \mathcal{C}$ for the (covariant) involution.
• A monoidal pseudofunctor $F : \mathcal{C} \to \mathsf{Cat}$.
• A family of "contraction" functors $F(A \otimes (X \otimes \bar{X})) \longrightarrow F(A)$, defined for all objects $A, X \in \mathcal{C}$, natural in $A$ and suitably laxly natural in $X$.

For example, you can take $\mathcal{C}$ to be the category of (not necessarily commutative) rings, with involution given by the formation of opposite rings. and $F$ to be the pseudofunctor assigning to every ring its category of modules. SImilarly with categories instead of rings and presheaves instead of modules. The idea is now that it should be possible to construct a compact closed bicategory, where the hom-category $A \to B$ is $F(A \otimes \bar{B})$, and horizontal composition is defined in terms of tensor followed by contraction. When applied to the previous two examples, I would expect to recover the bicategory of rings and bimodules as well as categories and profunctors, both of which are known to be compact closed bicategories.

So, what is the full list of structure and property that is needed in order to obtain such a general construction? Has this been worked out anywhere?

I imagine that somebody thinking about categorical network theory might have worked this out already, given that it's very much of that flavour: $F(X)$ could be the category of networks for a given type of ports $X$, and the contraction above corresponds to connecting up ports of the same type, though possibly "reversed" with one being input $X$ and the other output $\bar{B}$, or simply considered as "oriented" the other way.

John Baez (Nov 13 2023 at 09:05):

In network theory the monoidal bicategories we get, where the 2-morphisms are 'open networks' and composition is gluing the outputs of one network to the inputs of another, tend to have the property that the involution on object you're talking about is the identity.

John Baez (Nov 13 2023 at 09:05):

Thus in network theory I haven't been thinking about situations where that involution is not the identity.

Tobias Fritz (Nov 13 2023 at 09:07):

Hmm I see. I was thinking that it may be a bit like cobordisms, where one keeps track of an orientation, and that it might be e.g. the direction in which an electrical current is considered positive.

John Baez (Nov 13 2023 at 09:07):

The monoidal bicategories I've been studying decategorify to give, not just compact closed monoidal categories, but 'well-supported' compact closed categories, also known as 'hypergraph categories'. In such monoidal categories every object is a Frobenius monoid, and thus naturally its own dual.

John Baez (Nov 13 2023 at 09:08):

Yes, you might want to keep track of an orientation, e.g. to decide which direction of electrical current counts as positive. But so far we've been counting 'flowing into the inputs' as positive, as well as 'flowing out of the outputs'. This is an arbitrary choice, but it seems to work.

Tobias Fritz (Nov 13 2023 at 09:12):

Oh, but if you do have a distinction between inputs and outputs, then that's precisely the kind of situation that I'm describing: a network with input $A$ and output $B$ is an object in $F(A \otimes \bar{B})$.

Tobias Fritz (Nov 13 2023 at 09:15):

A network with only outputs of type $A$ and $B$ and no inputs would be an object in $F(\bar{A} \otimes \bar{B}) \cong F(\overline{A \otimes B})$, so that's different.

John Baez (Nov 13 2023 at 09:15):

Well, in any bicategory there's this distinction between inputs and outputs: the source of a 1-morphism is called its 'input', and the target is called its 'output'. That's all I meant by those words right now. I agree that you could think of things the way you're saying, but I haven't been talking about any $\overline{\phantom{X}}$ operation on objects.

I definitely like to think about symmetric monoidal bicategories of enriched categories, profunctors and natural transformations, which become compact closed bicategories via "op". Those include the case of algebras, bimodules and bimodule homomorphisms. So in those cases you have an explicit $\overline{\phantom{X}}$ operation on objects, namely "op".

But sadly, I haven't thought much about general machines to concoct compact closed bicategories generalizing this class of examples.

Tobias Fritz (Nov 13 2023 at 09:17):

I see -- so the examples of categorical network theory should still be an instance of what I'm suggesting, and often with a non-identity involution -- but it's just not an approach that has been considered in that context. Right?

John Baez (Nov 13 2023 at 09:41):

That sounds right, though I can't speak for all of categorical network theory! Decorated and structured cospan bicategories have an identity involution, just like plain old cospan bicategories.

Mike Shulman (Nov 13 2023 at 17:57):

I actually thought about this sort of thing myself, back when I was a graduate student. Many of the "hypergraph" cases are handled by my construction of a monoidal double category (and, thereby, a monoidal bicategory) from a monoidal fibration with cartesian base, and I looked at the same sort of examples you mention and thought there ought to be a version that works for them too. I started trying to write down some axioms but got bogged down in technicalities.

Mike Shulman (Nov 13 2023 at 17:58):

I expect it would be possible to push through the technicalities. However, now I think it ought to be just as easy to assemble this input data into a compact closed double category, and then lift the latter structure to a compact closed bicategory if you want. As John said, you can regard inputs and outputs as sources and targets, and then your "contraction" functors are just composition.

Tobias Fritz (Nov 13 2023 at 18:07):

Interesting, thanks! Oh well, in that case I'll have to try and make do without the most propitious general recipe.

Mike Shulman (Nov 13 2023 at 18:07):

Of course first one has to define a compact closed double category. This seems to be coming up a whole lot recently and should be done. @Christian Williams and I had some ideas about how to do it in the bifibrant case (i.e. proarrow equipments), which could then perhaps be translated to the general one, although most of the examples are bifibrant.

Tobias Fritz (Nov 13 2023 at 18:09):

Actually proving compact closure shouldn't be difficult in my case, the difficulty lies in getting a symmetric monoidal bicategory. Certainly building a symmetric monoidal double category first would be enough, since also the liftings seem to exist, but I have difficulty with constructing the horizontal composition of 2-cells. Often this is immediate from the universal property of composition of loose morphisms, but I haven't (yet) found such a universal property in my situation. I'll have to try harder then.

Tobias Fritz (Nov 13 2023 at 18:12):

I've already shown that the loose morphisms form a bicategory; the tricky case is composing two 2-cells where the intermediate tight morphisms is nontrivial.

Tobias Fritz (Nov 13 2023 at 18:13):

BTW if one uses the tight/loose morphism terminology, then what's the right phrase for "horizontal composition"? Would you say "loose composition"?

Mike Shulman (Nov 13 2023 at 18:52):

Yes, loose composition makes sense.

Mike Shulman (Nov 13 2023 at 18:53):

Would it be easier to show that you have a proarrow equipment in the sense of an identity-on-objects pseudofunctor $\mathcal{K} \to \mathcal{M}$, where $\mathcal{K}$ is the bicategory of tight morphisms and $\mathcal{M}$ the bicategory of loose ones? That way you don't have to talk explicitly about composition of 2-cells along a tight 1-cell.

Tobias Fritz (Nov 13 2023 at 19:08):

Yes and no. Actually that's the one thing that I've already done in detail, but using a different description of the loose morphisms (an equivalent bicategory). Let me call this description B, while the one I've been talking about previously is A. So the situation is this:

• Using B, I get a proarrow equipment in the sense of an identity-on-objects profunctor $\mathcal{K} \to \mathcal{M}$. In this picture, I might as well construct the double category directly; here the difference is relatively small. However, the problem with B is that the tensor product of loose morphisms is tricky to define.
• Using A, the tensor product of loose morphisms is pretty clear, but then I run into the difficulties with defining the loose composition of 2-cells. Constructing a proarrow equipment as an identity-on-objects profunctor $\mathcal{K} \to \mathcal{M}$ may be slightly easier (and seems worth considering further), but is not straightforward either because of the strongators. It's not at all obvious that $(g f)_*$ and $g_* \odot f_*$ will be isomorphic at all.

Tobias Fritz (Nov 13 2023 at 19:12):

Heck, might as well post what I have so far (Sections 4.6 and 4.7):
Wstar_bicategories.pdf
This just to give an impression and some context -- I don't think that this will be an easy read for anyone, especially at the stage of an incomplete draft with confusing notation. It's pretty similar to rings, ring homomorphisms and bimodules, with a lot of additional technical difficulties associated with the bimodules. Description A is the picture directly involving bimodules (Connes correspondences), while description B is what corresponds to A under the appropriate version of the Eilenberg-Watts theorem, which makes it a strict bicategory.

Tobias Fritz (Nov 13 2023 at 19:36):

Perhaps the right way to proceed is simply to understand the equivalence of A and B better and then develop both the equipment and the monoidal structure in parallel.

Tobias Fritz (Nov 13 2023 at 19:38):

Has anyone worked out what a definition of symmetric monoidal structure on an equipment in the pseudofunctor $\mathcal{K} \to \mathcal{M}$ sense would be, such that the resulting gadgets are equivalent to fibrant monoidal double categories, and such that I wouldn't have to make $\mathcal{M}$ into a symmetric monoidal bicategory from the start?

Evan Patterson (Nov 13 2023 at 19:59):

I don't know much about $W^*$-algebras so this might be totally off base, but here's an idea anyway. Your notion of tensor product (the "Deligne tensor product") seems to be based on the usual universal property classifying bilinear maps. So what if you defined a double multicategory (a pseudocategory in the 2-category of multicategories) of $W^*$-categories, $W^*$-functors, $W^*$-bimodules, and $W^*$-transformations. You would then show that this double multicategory is representable and so obtain a monoidal double category without grinding through a lot of coherence calculations. Finally, from that you could presumably extract a monoidal bicategory, although for me the double category would really be the fundamental thing.

Tobias Fritz (Nov 13 2023 at 21:01):

Wow! This is something that I hadn't thought of yet and it sounds worth taking seriously. I don't really know what a "pseudocategory in the 2-category of multicategories" amounts to yet, but I see a very similar definition in Multivariable adjunctions and mates, which also looks similar in spirit to my situation. So I'll give it some thought and will report back. Thanks @Evan Patterson !

Christian Williams (Nov 13 2023 at 21:09):

To me, it sounds like it might be harder to characterize the construction of $W^\ast$-categories "from scratch". You're starting from Banach-enriched categories, right? Those should form a compact closed equipment, by the "monoidal bifibration" and then "monads and bimodules" constructions. Then maybe it could be easier to consider what construction is being applied to $\mathrm{BanCat}$?

Tobias Fritz (Nov 13 2023 at 21:20):

That's true, but there are a lot of features specific to $W^*$-category theory that set it apart from mere Ban-enriched category theory, to the extent that the latter has little bearing on the former. For example, the $W^*$-version of presheaves are "Hilbert presheaves", which are particular Ban-presheaves equipped with an inner product in the category they're defined on. This is not something that an ordinary Ban-presheaf has. And one of the simplifying features of the theory is that every $W^*$-functor preserves $W^*$-limits -- also something that one would not expect from general enriched functors. Finally, the norms on the hom-spaces on a $W^*$-category are actually uniquely determined by the other structure! So all in all, the comparison with Ban-enriched categories is unfortunately not very helpful.

Christian Williams (Nov 13 2023 at 21:32):

Yes, but if you knew a construction (on double categories with some structure) which when applied to $\mathrm{BanCat}$ gives $W^\ast \mathrm{Cat}$, then you could derive the structure and properties of $W^\ast \mathrm{Cat}$ from that of $\mathrm{BanCat}$ and that of the construction. Right? Maybe it's not what you want to do, but it seems like an option (unless I'm missing something).

Evan Patterson (Nov 13 2023 at 21:56):

Thanks @Tobias Fritz, it would be cool if that actually works out! Yes, that paper by Cheng, Riehl, and Gurski is the only one I know of that uses double multicategories, specifically strict double multicategories which are category objects in the category of multicategories. (Shulman also has work using the related idea of a double polycategory.) For anyone following along, Cheng et al have a nice picture of what a cell in a double multicategory looks like:

image.png

Anyway, I've been toying around with double multicategories, not much further than having a few examples that I like, and I think they are a natural idea that deserves to be studied further.

Mike Shulman (Nov 13 2023 at 23:32):

I prefer to call these "multi double categories", since "double multicategory" sounds to me like something that is multi in two directions, whereas these are only multi in one direction.

Evan Patterson (Nov 13 2023 at 23:39):

I see your point, and "multi double category" has the advantage of being parallel with "monoidal double category." I guess the idea behind "double multicategory" is that a "double gadget" should be a "pseudocategory in the 2-category of gadgets."

Mike Shulman (Nov 13 2023 at 23:42):

Yeah... but I don't like that idea. A "double X" should involve two Xs, not an X together with a category.

Evan Patterson (Nov 13 2023 at 23:43):

On that idea, a "double X" does involve two Xs, an X of objects and an X of morphisms!

Mike Shulman (Nov 13 2023 at 23:44):

Um. That's never been my interpretation of the phrase "double category" -- I always considered it as referring to the vertical category and horizontal category that share their objects.

Evan Patterson (Nov 13 2023 at 23:46):

Fair enough. I guess I've just so internalized the picture of a double category as a pseudocategory in Cat than I now want to think of the "doubling" as being the two categories, but you must be right that that was not the original idea behind a double category.

John Baez (Nov 13 2023 at 23:59):

Nice double entendre with "internalized" there. And of course you would be going for a double entendre.

Mike Shulman (Nov 14 2023 at 00:03):

Actually I have no idea whether that was the original intuition behind the phrase "double category". But I think my interpretation is borne out by the generalization to "triple category", which when interpreted as an internal double category in Cat consists of four categories rather than three, whereas it does have three directions of arrows.

Tobias Fritz (Nov 14 2023 at 08:22):

Christian Williams said:

Yes, but if you knew a construction (on double categories with some structure) which when applied to $\mathrm{BanCat}$ gives $W^\ast \mathrm{Cat}$, then you could derive the structure and properties of $W^\ast \mathrm{Cat}$ from that of $\mathrm{BanCat}$ and that of the construction. Right? Maybe it's not what you want to do, but it seems like an option (unless I'm missing something).

Got it. That's a nice idea, but I don't know of any general categorical construction that would give $W^\ast \mathrm{Cat}$ from $\mathrm{BanCat}$. I can't claim that it doesn't exist since I wouldn't know how to make this precise. But finding one would be like finding a general construction construction that gives the category of Hilbert spaces from the category of Banach spaces, which seems implausible.