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I have two questions related to Selinger: A survey of graphical languages for monoidal categories.

1) On page 60 there is a summary of some monoidal categories with extra structure. Among them are the well known braided and symmetric versions. Due to the periodic table, these two monoidal categories stem from 3- resp. 4-categories. Can some of the other special monoidal categories also be deduced from certain higher categories?

2) There is another type of string diagrams not present on page 60. Namely you can do the following. Assume you have a string diagram of a monoidal category with no extra structure, i.e. a string diagram on a plane square. Bend such a square to a cylinder by identifying the left and right side of that square. The diagram is now "painted" on a cylinder und you can also compose cylinders (but only "vertically"). Does this correspond also to some extra structure on a monoidal category? In fact, this extra structure should somehow reduce the dimension of the monoidal category by one, because tensor expressions are cyclically isomorphic, e.g. the objects $A_1\otimes A_2\otimes A_3$, $A_2\otimes A_3\otimes A_1$, $A_3\otimes A_1\otimes A_2$ are coherently isomorphic. This corresponds to the top and bottom of the cylindrical diagrams being a circle.

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2 Answers

up vote 6 down vote accepted

As you point out, in such a situation it may not be possible to compose horizontally, hence in full generality you cannot assume your underlying category to be monoidal. On the other hand, you still want to keep track somehow of the tensor product. I think it can be handled as follow:

The structure you are looking for should be the data of a (strict, say) monoidal category $C$, an arbitrary category $D$, a functor $F:C\rightarrow D$ and a natural isomorphism $$\gamma_{-,-}:F(-\otimes -)\rightarrow F(-\otimes^{op} -)$$. It seems to me that one natural coherence condition to impose is: $$\gamma_{Z\otimes X,Y}\circ\gamma_{X\otimes Y,Z} =\gamma_{X,Y\otimes Z}^{-1}$$

Maybe you need the "reversed" condition as well.

Anyway, examples of such categories may be constructed as follows: let $C$ be a rigid monoidal category enriched over some category $D$, let $F$ be the functor $Hom_C(1,-)$ and $\gamma$ be given by $$Hom_C(1,X\otimes Y)\cong Hom_C(X^*,Y)\cong Hom_C(1,Y\otimes X)$$

I'm not aware of such a definition in the litterature, but the obviously related notion of a categorical structure involving braid diagrams drawn on a surface appears in this paper by Calaque-Enriquez-Etingof for the torus (under the name elliptic structure) and for abritrary surfaces in Philippe Humbert's thesis.

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2) You may be interested in this paper.

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