6
$\begingroup$

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal product) and the tensor product of bimodules. I haven't seen any diagrammatic calculus that allows for this mixing, but I haven't searched exhaustively. It seems that if I do something naive I could get something completely wrong.

Has anybody used/seen such a thing?

$\endgroup$
3
  • $\begingroup$ Does there exist a fiber functor from your category of bimodules to the underlying category? $\endgroup$ Jul 28, 2014 at 9:22
  • $\begingroup$ I'm not sure, how would that help? $\endgroup$
    – David Roberts
    Jul 28, 2014 at 9:49
  • $\begingroup$ In this case, one could draw pictures like e.g. in Majid: Algebras and Hopf Algebras in Braided Monoidal Categories (Section 2.3, first picture, the subpicture in the dotted square). (Just treat the braiding as symmetric, and do the same for the right action as well). $\endgroup$ Jul 28, 2014 at 10:13

2 Answers 2

5
$\begingroup$

In general, the correct generalization of the "stringy" notation for bicategories — in which objects label regions in $\mathbb R^2$, 1-morphisms label codimension-1 defects (whose projection to $\mathbb R$ has no critical points), and 2-morphisms label codimension-2 defects, and composition is a "pushforward" of defects — to monoidal bicategories is to put everybody in $\mathbb R^3$ (giving no label to the ambient 3-space, and letting the 2-dimensional regions wiggle around such that they are parameterized by their projections to $\mathbb R^2$). If you want braided monoidal bicategories, you should use $\mathbb R^4$ (but no 3-dimensional defects), and if you want sylleptic monoidal, you use $\mathbb R^5$. In bicategories, if you go to $\mathbb R^6$, you already get symmetric monoidal, but more general $(\infty,2)$-categories do not, I think.

In general, the "true" topological setting for symmetric monoidal bicategories is of surfaces-with-defects that are not embedded into any $\mathbb R^n$, or (better?) that are embedded into $\mathbb R^\infty$. Again you should include some framing data to keep the surfaces from swinging around on themselves. I think it works to give every 2-dimensional region an embedding as a domain in $\mathbb R^2$, such that restricting the 2-dimensional parameterization to any 1-dimensional boundary component commutes (up to specified homotopy) with embedding in $\mathbb R^2$ and then projecting to $\mathbb R$.

I'm not aware of any paper that writes these pictures up carefully and proves coherence theorems comparable to Ross–Street for plane string diagrams, although that doesn't mean there aren't any. Lurie's TQFT paper contains essentially all of these ideas, but it's not the focus — but he does address the tangle hypothesis in the last section. The consensus among people I know who've studied the paper (I am not one of them) is that it is complete and correct: what Lurie calls an "outline" most people would call a "detailed proof". (See for example Section 2 of Scheimbauer's thesis, where Lurie's definition of the bordism category is filled in and all facts proved; but I don't think Scheimbauer has posted her thesis publicly yet.)

For monoidal (rather than symmetric monoidal) bicategories, you can use Gray categories: a monoidal bicategory is a tricategory with one object, and Gray proved a coherence result that every tricategory is weakly equivalent to a Gray category. (But be careful with functors.) I haven't read Hummon's thesis, but it seems to provide such "surface diagrams" for arbitrary Gray categories. See also arXiv:1211.0529 and the n-lab page on Surface Diagrams.

$\endgroup$
4
  • 1
    $\begingroup$ I second the recommendation to look at the arXiv article that Theo links to at the end, which provides a surface diagram calculus for Gray-categories with duals, parallel to the Joyal-Street calculus. $\endgroup$
    – Todd Trimble
    Jul 28, 2014 at 17:44
  • $\begingroup$ Schaumann's thesis here also includes a bunch of information on doing graphical calculus for module categories. $\endgroup$ Jul 28, 2014 at 18:55
  • $\begingroup$ Good to know Lurie's paper has been given some approving nods. By Ross--Street, do you mean Joyal--Street? ;-) I may just have to do movie diagrams in the end. And all this is to check whether something is a strong/lax monoidal 2-functor with values in said symmetric monoidal bicat :-) $\endgroup$
    – David Roberts
    Jul 28, 2014 at 23:28
  • $\begingroup$ @DavidRoberts: Sorry for the slow response. Yes, of course Joyal--Street --- I clearly wasn't thinking straight, and went "Street was one of the names ... what's the name that goes with Street?" $\endgroup$ Aug 2, 2014 at 1:48
2
$\begingroup$

I was doing something like that when I was working on my PhD thesis. See string diagrams in the end of this paper http://www.tac.mta.ca/tac/volumes/25/1/25-01abs.html

The idea is to put strings into a box when you want to take a tensor product of bimodules. This paper works with comodules, but with modules it will be the same.

I didn't quite try to make this notation completely precise, but I was finding it very useful for calculations. Hope it helps.

$\endgroup$
1
  • 2
    $\begingroup$ Ah, that equation (32) on page 30 is a good thing to think about. Thanks! $\endgroup$
    – David Roberts
    Jul 28, 2014 at 23:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.