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? 



String diagrams for bimodules over noncommutative algebras?
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](http://escholarship.org/uc/item/5b24s9cc), but it seems to provide such "surface diagrams" for arbitrary Gray categories. See also [arXiv:1211.0529](http://arxiv.org/abs/1211.0529) and the [n-lab page on Surface Diagrams](http://ncatlab.org/toddtrimble/published/Surface+diagrams).


