I just saw this definition/interpretation of monoidal categories and I have two questions about it.
The idea is to consider the objects of the monoidal category as 1-cells and the tensor product as 2-cells.
- Where are the morphisms of the monoidal category?
Thinking of $\mathcal{Cat}$ as bicategory, string diagrams in it draw 0-cells as two dimensional spaces, 1-cells as one dimensional strings between the spaces, and 2-cells as zero dimensional points on the strings.
But I don't see how this is the same as with string diagrams for monoidal categories ---if we want to think of them as bicategories with one 0-cell---, where objects are strings and morphisms are dots. (There may be different two dimensional spaces but only one 0-cell.)
- What is happening here? Is there a more general way to interpret string diagrams that encompass both ways or is there actually a way to think of string diagrams for monoidal categories as string diagrams for bicategories with one object? Or just how should I think about it?
Let's call the potential interpretation of string diagrams for bicategories I gave above bi-string diagrams, and the interpretation of string diagrams for monoidal categories I gave above tensor-string diagrams.
Being explicit, a bi-string diagram is drawing 0-cells as spaces, 1-cells as strings and 2-cells as dots; wheareas a tensor-string diagram is drawing objects as strings and morphisms as dots.
I must note that I assumed that bi-string diagrams are drawn this way by analogy with $\mathcal {Cat}$, but I'm not sure it's actually this way in general.
I do realize that bi-string diagrams for $\mathcal{Cat}$ can also be interpreted as tensor-string diagrams for the functor category with horizontal composition as tensor product. (Hence the name horizontal.) So it is not clear which interpretation (bi- or tensor-) is more general.
I'm a bit confused with all of this so I might be making some profound mistakes, and so I'm sorry if that's the case.
Thanks in advance for any help!