Order of factors for unit/counit of duals

Question

While working through Categorical quantum mechanics by Abramsky and Coecke, I noticed that their definition of the order of factors for the unit and counit of a dualizable object disagree with the one at the nlab. They cite the 1980 Kelly and Laplaza paper on compact closed categories for their definition, but this paper defines the order of the factors the same way the nlab does. They seem to use their ordering for the remainder of the paper (i.e. defining $f^*$), which makes me wonder

What impact does the order of factors have on the contents of their paper? Is there a reason they chose to reverse the ordering?

Right off the bat we end up with the fact that the counit doesn't technically embed the complex numbers on the diagonal of a matrix, and that bra-ket notation looks backwards. Are there any more serious one?

Answer 1

Just to air out some thoughts here:

• The thing I reach for when deciding the order to write here is this: Let $X$ be an object and $[X , I]$ its internal hom to the unit. Note that $X = [I, X]$ is the same as the internal hom from the unit. If $X$ is dualizable, then it's natural to write the counit of the dualizability data so that it agrees with the evaluation map.

• But how do we write the evaluation map? The evalutation map is an internalization of composition. So the way you write the evaluation map is going to depend on whether you're writing composition in diagrammatic order, or in op-diagrammatic (or "conventional") order.

I think I tend to instinctively write the counit as $X \otimes [X , I] \to I$ because I'm thinking of it as evaluation written in diagrammatic order. Probably the reason this is backwards from Dirac's bra-ket notation is that bra-ket corresponds to thinking of evaluation as written in op-diagrammatic order -- the order you get when you write $g(f(x)) = (g \circ f)(x)$.

So this is another case where there might be a distinction to be drawn between the order that you write things on paper and the order of "priority". Some people -- such as reverse Poles write in op-diagrammatic order $(g \circ f)(x) = f(g(x))$. Are they using a different notion of composition, or rather the same notion of composition written differently? I'm not sure!

The discussion of "chirality" in 1.1.4 - 1.1.5 of Berrick and Keating's Rings and Categories of Modules seems relevant here. It would be nice if we had better ways of talking about this sort of issue -- maybe Berrick and Keating's terminology should be more widely adopted!