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Stream: learning: questions

Topic: Braidings in symmetric monoidal double categories

Evan Patterson (Oct 23 2020 at 22:19):

A double category is succinctly defined as a category $S, T: D_1 \to D_0$ internal to $\mathsf{Cat}$, where by convention the morphisms of $D_0$ are the vertical 1-cells and the objects of $D_1$ are the horizontal 1-cells. The asymmetry in this definition is only apparent, because the horizontal and vertical directions turn out to be structurally the same. Similarly, the definition of a monoidal double category looks asymmetric but is actually not. For simplicity, let's assume everything is strict.

A braided monoidal double category is defined to be a monoidal double category such that (i) $D_0, D_1$ are braided monoidal categories, (ii) $S,T$ are strict braided monoidal functors, and (iii) the braiding 2-cells are a transformations of double categories. When I unpack this data, what I see is that for every pair of objects $A,B$, there is a vertical braiding 1-cell $\sigma_{A,B} = A \otimes B \to B \otimes A$, and for every pair of horizontal 1-cells $M: A \to C$ and $N: B \to D$, there is a braiding 2-cell $\sigma_{M,N}$ whose top and bottom 1-cells are $M \otimes N$ and $N \otimes M$ and whose left and right 1-cells are $\sigma_{A,B}$ and $\sigma_{C,D}$.

This setup seems truly asymmetric to me, since there are braiding 2-cells in the horizontal direction but not the vertical one. Do I have this right?

John Baez (Oct 24 2020 at 16:28):

Yes, the setup is asymmetric and this is intentional. I was confused for a while when you said "2-cell in the horizontal direction", since of course a 2-cell is a square, not going in any direction in particular. But anyway, the setup is asymmetric.

John Baez (Oct 24 2020 at 16:31):

A good example to keep in mind is the symmetric monoidal double category of spans of sets. Here the objects are sets, the vertical morphisms are functions and the horizontal 1-cells are spans of sets. We mainly want functions

$\sigma_{A,B}: A \times B \to B \times A$

not spans of this sort. And that's what the definition of symmetric monoidal double category provides. The 2-cells sort of go along for the ride.

John Baez (Oct 24 2020 at 16:32):

But this particular double category is "fibrant", meaning that any function can be turned into a span in two ways. As a consequence of that we also get spans corresponding to the functions $\sigma_{A,B}$.

John Baez (Oct 24 2020 at 16:34):

More generally, this will happen whenever your symmetric monoidal double category is "isofibrant", meaning that any vertical isomorphism can be turned into a horizontal 1-cell in two ways, obeying some equations. Structured cospan double categories are all isofibrant!

Evan Patterson (Oct 24 2020 at 18:55):

Thanks, I'm glad to know I'm not going crazy! The intuition I had developed about double categories was that "the definition is asymmetric w.r.t. horizontal and vertical but you should think about it symmetrically," so the definition of braided double category threw me off. Your explanation and the example of spans helps, thanks.

John Baez (Oct 24 2020 at 19:09):

I don't really think you should think of the double categories Shulman and I are using - pseudo double categories - symmetrically. They're not. The older "strict" double categories are.

John Baez (Oct 24 2020 at 19:10):

That is, the definition of a strict double category is symmetric under interchanging vertical and horizontal, even though an individual instance of a strict double category might not have this symmetry.

John Baez (Oct 24 2020 at 19:14):

The definition of a pseudo double category is asymmetric since composition of vertical 1-morphisms is strictly associative - as suitable for functions - while the composition of horizontal 1-cells is associative only up to isomorphism - as suitable for spans, or cospans.

John Baez (Oct 24 2020 at 19:15):

(You probably know all this, but I want the world to know, and it sometimes it's reassuring to hear things you know.)