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Is there a kind of string diagram that works for closed (not compact) monoidal categories? I feel like I’ve seen this somewhere before but I can’t find it or reconstruct it. There is the feeling that you can “turn around” strings like in a compact-closed category, but the turning around has to be somehow relative to some other thing that you’re homming into.
Is this related to the ’popping the bubble’ that John Baez talked about when discussing lambda calculus? It was a sort of virtual ’turning around’ of strings.
I don’t know, can you give me a link?
These notes from John’s QG seminar, Autumn 2006 (or Fall, for those in Amerika). See also the blog entry.
That’s what I was looking for, thanks!
Has anyone ever made that notation precise? I.e. specified exactly what you can and can’t do with the bubbles?
The successive weeks I think expound the rules a bit more just along the way. If you go to the appropriate page of the QG seminar, and look under the classical vs quantum computation section, there may be something useful. I recall that there was some more somewhere, but I can’t remember if it was on the blog or in the next session of the seminar. Certainly, putting the this diagrammatic calculus up on the lab would be a great idea.
A little bit here about categorifying (rewriting as a surface between string diagrams) the setup. Actually there is not as much at the seminar pages as I thought. It must be somewhere else.
Can you recommend something of a dry/compressed/technical introduction to string diagrams, in particular, for the more-or-less standard variant?
I will not belabour here what I mean by “more-or-less standard variant”. But it should
go considerably further than only discuss the usual application to adjunctions in 1-categories
be aware of bicategories
string diagram itself is currently not such a kind of reference.
EDIT: having had another go at searching the literature, I would say that, up until here (the requirement below, to the effect that this should be a detailed reference work that one can quote from, seems to amount to too tall an order for there to be such an exposition yet, but it seems useful to recommend Schommer-Pries’ introduction here), the best “fit” to the “bill” above is:
C. Schommer-Pries, Section A.4 of “The Classification of Two-Dimensional Extended Topological Field Theories”, arXiv:1112.1000v2
The above section is a reasonably self-contained, bicategorical, illustrated introduction, informally discussing the main issues.
END of EDIT.
Actually, “introduction” above is not the right word, since I know the basics, and am rather looking for something like reference to work with and make references to.
I had a close look at Hotz’s dissertation work, which seems rigorous, but there, no bicategories were around yet, and there is translation overhead involved (not so much German->English, but (terminology back then)->(terminology now).
I also had a close look at Joyal and Street’s Adv. Math. articles, but these are in the style of research papers, with good reason, wanting to reach certain heights, but do not serve as references to consult when one has some problem with string diagrams to deal with.
EDIT: And Joyal and Street do not treat bicategories.
I am fine with being told that currently there is no such article. I am not asking for beign sent many references, nor for someone to write an introduction to string diagrams here, this would perhaps dublicate too much of what is available (given an internet connection).
Maybe some work which makes a conscious effort to be formal and systematizing, such as Leinster’s monumental two-pages-per-definition article?
Joyal and Street explicitly developed string diagrams for (flavors of) monoidal categories, but it should be immediately pointed out that these are strict monoidal categories where associativity and unit isomorphisms are identities. Indeed they spend some time at the beginning of their article discussing monoidal strictification, before getting into details of the geometry.
In the same way, string diagrams can be interpreted or evaluated in bicategories, but only strict ones which are usually called 2-categories. Here, if a string diagram is reckoned as a stratified topological space
then the connected components of are interpreted or evaluated as objects of the 2-category. My memory is that this extension is discussed in Street’s article Categorical Structures.
In the same way, string diagrams can be interpreted or evaluated in bicategories, but only strict ones which are usually called 2-categories.
The “only strict […]” is unlikely to be literally true (and possibly you did not mean to make such a strong claim), for example because Garner–Shulman in Adv. Math. 289 make extensive and essential use of a string-diagram-calculus in weak 2-categories.
I am studying this article very closely, and was looking for something more or less exactly matching Garner–Shulman Adv Math. Asking Mike Shulman directly appears not an option since most of the questions I have are not (yet) of the form one can conveniently ask, and since most of them are of one of the myriad kinds of questions that one should clear up for oneself instead of bothering authors with.
To get back to your comment: did you have something particular in mind when saying that string diagrams cannot be interpreted in weak 2-categories? If there is something essential to watch out for, this would be very relevant for what I am currently doing, and would appreciate to be told about it (or referred to a relevant place in the literature).
You should direct your questions to Mike then.
Re #11: perhaps, if askable question have formed, maybe.
this would be very relevant for what I am currently doing
What are you doing, if I may ask?
There is a part of combinatorics, and a well-known open problem, where distributors come naturally, sometimes it feels as if they were made for it. This has not been observed, and I am working on a theory and a proof.
Again, distributors in the most basic sense come very naturally in this, but it is very debatable whether one should make the effort to learn technology as advanced as the one developed in e.g. Garner–Shulman.
Personally, I think it worth the while and have decided to do so. I like not to treat structures as black-boxes, or be only a consumer of other parts of mathematics. This slows things down of course.
For various reasons I would prefer not to give more details, at least not here and now and in a few words.
I commend you for that attitude. Too many mathematicians use black boxes as it is (sometimes hard to avoid, but I agree one should fight the good fight).
One can interpret string diagrams in non-strict bicategories too. One way to do it is to invoke the coherence theorem for bicategories, interpret the string diagrams in a strict 2-category, and then transfer across the coherence equivalence. But a more pleasing way to do it (though I’m not sure whether this is written down anywhere) is to regard the interpretation of string diagrams as a way to prove the coherence theorem for bicategories: the string diagrams themselves form a strict 2-category, and the theorem about their interpretation proves that this is nevertheless the free bicategory on the input data.
Re #16: that’s pretty much what I expected. Perhaps related is Power’s theorem on interpreting pasting diagrams in bicategories (I think it’s recalled in Verity’s thesis).
Re #16: How exactly do string diagrams form a strict 2-category? I would have thought that the diagrams would have to be rescaled every time you glued them together, and then the diagram of would be different from that of .
@Oscar diagrams are to be considered up to ambient isotopy, I believe.
Right. If I remember correctly, if you don’t quotient by isotopy, then you may get some kind of weak 2-category, but it’s neither the biased sort of bicategory that people usually use nor the unbiased sort that arises naturally from 2-monad theory, so I don’t think it’s a whole lot of use.
On top of that, if you don’t quotient by isotopy, then you lose the middle-4 interchange equations (the ability to smoothly handle such interchanges is really the basic reason string diagrams are so effective).
Right, thanks Todd. I think what I was thinking about is quotienting only by isotopy rel boundary, so that you would still have interchange inside the diagrams but concatenation of the boundaries (1-cells) would not be strictly associative.
Re #17:
Perhaps related is Power’s theorem on interpreting pasting diagrams in bicategories (I think it’s recalled in Verity’s thesis)
Yes it is. I added relevant context in a thead on string diagrams, since the present one is about string diagrams.
Terminological/linguistic comments on parts of the above discussion (I think all of this to be evidently true, and am adding this mostly for inexperienced readers and for the possibility that someone would like to say a thing or two on this, but do not expect answers to this):
And just to give another reading-advice tip to people not very experienced in this:
in the literature, it sometimes helps to replace the word “data” for “element of the model”, especially if you have some model-theoretic background, and, more generally, think of “formulas-versus-models”.
More extensively, to give an example, there are more than one discussions in the literature revolving around the fact that
The reason is of course that in the case of strict equality, there is essentially no further _information attached to a particular equality of two things, the logical judgement that the two are equal is the long and the short of it, while in the case of equality up to coherent specified morphisms, each equality gets attached further information/data to it.)
One briefer way and more usual way of rephrasing this:
Re #25: while some authors make this axiom-datum-distinction, it seems rather illusory.
This is not a difference in kind, only in degree. Both the concept of monoidal category and monoidal bicategory have (0) axioms and (1) models.
Both for monoidal categories, and for monoidal bicategories, there is a pentagon-axiom saying that there be a pentagonator. It is only that for the former, there tends to be less information in models of the pentagon axiom.
The only thing I think I would take issue with is the claim that “there be a pentagonator” is an axiom. Usually in first-order logic an “axiom” refers to a logical statement that can only be true or false, but the pentagonator in a monoidal bicategory really is data; it matter what it is and not merely that it exists.
In fact, in (homotopy) type theory and higher category theory, it is generally more fruitful to go the other way and regard “axioms” as a degenerate special case of data. Cf. for instance negative thinking.
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