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Let $A$ be a ring, and $L,M,N$ an $A^\mathrm{op} \otimes A$-modules. $L \otimes_A M$ is then an $A \otimes A^\mathrm{op}$ module so we can tensor it again with $N$ to get an abelian group $(L \otimes_A M) \otimes_{A \otimes A^\mathrm{op}} N$.

This is basically tensoring $L, M, N$ in a triangle, in a cyclic fashion. I am under the impression that this idea is useful in understanding Hochschild (co)homology, but I don't actually know anything about it unfortunately.

My question is that given that we could form tensor products of more complicated graphs than a cyclic triangle, even varying $A$, or having multimodules instead of bimodules, does "tensoring diagrams of multimodules" exist in the literature?

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Consider a bicategory $\mathrm{Mod}$ whose objects are rings $A$, $B$..., whose morphisms $M : A \rightarrow B$ are $A$-$B$ bimodules (which are the same as $B^\mathrm{op}\otimes A$-modules), and whose 2-cells are bimodule maps. Composite of bimodules $M : A \rightarrow B$ and $M : B \rightarrow C$ is defined by $N\circ M = N\otimes_B M$.

$\mathrm{Mod}$ is a monoidal bicategory. The tensor product of objects is the tensor product of rings. The monoidal unit $I$ is the ring of integers.

$A^\mathrm{op}$ is the bidual of $A$ in $\mathrm{Mod}$. The unit $I \rightarrow A^\mathrm{op}\otimes A$ and the counit $A^\mathrm{op}\otimes A \rightarrow I$ both are $A$ considered as a module.

Compositions in $\mathrm{Mod}$ correspond to various tensorings of modules. While, to represent compositions in $\mathrm{Mod}$ one can use the graphical calculus for monoidal categories with duals.

Thus, given modules $L, M, N : A \rightarrow A$, the tensor product in the OP can be graphically represented as:

enter image description here

More precisely, this string diagram represents the morphism $I \rightarrow I$ of $\mathrm{Mod}$ which is the tensor product in the OP.

One can draw various such string diagrams involving modules, which will correspond to different kinds of tensorings. "Multimodules" can be represented by a labeled nod which has few string coming in and going out from it, labeled by rings, which can indeed vary.

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  • $\begingroup$ See Appendix A of my paper arxiv.org/pdf/1409.8672v1.pdf for the corresponding construction (multifusion of multimodules) in the worlds of von Neumann algberas. In that case, not all graphs that are allowable (see e.g. Warning A.6 of my paper), and it is a very interesting matter to figure out which ones are allowable and which ones are not. The main difference with pure algebra is that with von Neumann algebras bot every $A$-$A$-bimodule is an $A\otimes A^{op}$-module. $\endgroup$ Nov 1, 2014 at 21:41

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