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Jules Hedges (Feb 02 2021 at 10:36):Cole Comfort said:
They go back much earlier, like in this paper of Cockett and Seely. But I don't know who invented them, or if they are just folklore.
Huh, I believed that Aleks Kissinger invented functorial boxes about 10 years after that paper, so much for that
Amar Hadzihasanovic (Feb 02 2021 at 10:54):I learnt about them from this paper by Paul-André Melliès who credits Cockett, Seely, and Blute with their invention.
Amar Hadzihasanovic (Feb 02 2021 at 10:55):@Jules Hedges are you sure you are not thinking of Aleks's !-boxes?
Cole Comfort (Feb 02 2021 at 11:05):Can !-boxes be seen as functor boxes for monoidal (co)monads?
Cole Comfort (Feb 02 2021 at 11:06):Or are they completely unrelated, I wonder
Jules Hedges (Feb 02 2021 at 13:58):Amar Hadzihasanovic said:
Jules Hedges are you sure you are not thinking of Aleks's !-boxes?
I'm sure I was thinking of them. I guess !-boxes are a special case of functorial boxes
Amar Hadzihasanovic (Feb 02 2021 at 14:13):I think there may be a translation between the two formalisms (where a diagram with !-boxes becomes a “diagram with holes”, aka a context, and the holes are inside functorial boxes) but it's not immediate...
For example an instantiation of a !-box can attribute a different type to nodes that are outside the !-box, which a functorial box can never do.
Perhaps we can ask @Aleks Kissinger if he tried to make a connection.
Amar Hadzihasanovic (Feb 02 2021 at 15:48):(I split the topic too since this is no longer about chemistry)
Aleks Kissinger (Feb 03 2021 at 07:53):a long time ago, Ross Duncan and I were looking for some kind of categorical interpretation of !-boxes as comonads or similar, but never got anywhere. as Amar said, the challenge is instantiating them within a diagram changes the types of morphisms around them, which is very much not functor-like. We ultimately settled on them being described purely as syntax, living at the level of graphs
Aleks Kissinger (Feb 03 2021 at 07:57):the earliest i've seen something like functorial box (or "functorial bubble") notation is in the appendix of Spinors and Spacetime (Penrose and Rindler, 1984). the authors introduce string diagram notations for various differential operations on tensors by drawing different kinds of bubbles around them
Aleks Kissinger (Feb 03 2021 at 08:01):however, they don't insist that composition (aka tensor contraction) is preserved, so they are an instance of something more general than a functorial box. also, many times there are different numbers of wires on the outside of a box than the inside. for instance, taking the gradient of a scalar field yields a vector field. this is pictured as a box with no wires inside and 1 wire outside