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Stream: learning: questions

Topic: operads and parameterized categories

Kyle Wilkinson (Jul 28 2023 at 20:49):

As I understand it, a parameterization or para construction of a monoidal category C given as Para(C) inherits its own monoidal category structure from C, per the nLab (though the section is not written up yet to explain exactly how that is). Also, as I understand it, one can construct an underlying operad from a monoidal category in general, so then it should be possible to construct an operad around a parameterized monoidal category? If so, has there been work in this direction? Since I think it should be possible, I have been looking for it, but haven't found anything yet.

Matteo Capucci (he/him) (Jul 31 2023 at 12:48):

It can in the same way you get one from any monoidal category: the operations $X_1, \ldots, X_n \to Y$ are given by morphisms in $\bf Para(\cal c)$ of type $X_1 \otimes \cdots \otimes X_n \to Y$

Cole Comfort (Jul 31 2023 at 13:00):

Matteo Capucci (he/him) said:

It can in the same way you get one from any monoidal category: the operations $X_1, \ldots, X_n \to Y$ are given by morphisms in $\bf Para(\cal c)$ of type $X_1 \otimes \cdots \otimes X_n \to Y$

Yeah every monoidal category can be regarded as an operad; but you could even go further and define the Para construction as a monad on operads instead of on monoidal categories.... so that given an operad, it produces a parametrized operad.

I'm not sure of any intersting examples, but maybe it is worth noting at least that this can be done (unlike say for optics).

Matteo Capucci (he/him) (Jul 31 2023 at 13:06):

At last CT conference Eugenia Cheng gave a talk where she mentioned a kind of operads whose operations are parameterised by another operad... That's the most direct thing I can think of, though it wouldn't be much different than the above. I'd love to know more myself about these parametric operad bc they sound cool but I don't know where to look.

Kyle Wilkinson (Jul 31 2023 at 13:26):

Matteo Capucci (he/him) said:

It can in the same way you get one from any monoidal category: the operations $X_1, \ldots, X_n \to Y$ are given by morphisms in $\bf Para(\cal c)$ of type $X_1 \otimes \cdots \otimes X_n \to Y$

Thanks, yes that's what I thought, but was surprised to not see it in any application yet, since I think it would be useful.

Matteo Capucci (he/him) (Jul 31 2023 at 14:28):

it's just that apparently the ACT community prefers symmetric monoidal cats to coloured operads, but the two things should be interchangeable

Matteo Capucci (he/him) (Jul 31 2023 at 14:28):

On a different note, $\bf Para(\cal C)$ has a 2-dimensional structure I haven't seen in operads

Cole Comfort (Jul 31 2023 at 14:37):

Matteo Capucci (he/him) said:

On a different note, $\bf Para(\cal C)$ has a 2-dimensional structure I haven't seen in operads

There is a notion of 2-multicategories (essentially 2-operads), so I imagine that it would result in the same sort of thing.

Also coloured operads and monoidal categoires are not interchangable (although there is an adjunction between the category of coloured operads and monoidal categories). Monoidal categories are precisely the representable coloured operads; in a coloured operad you can't cut two circuits together along more than one wire at once

Kyle Wilkinson (Jul 31 2023 at 14:39):

Matteo Capucci (he/him) said:

On a different note, $\bf Para(\cal C)$ has a 2-dimensional structure I haven't seen in operads

From what I see, operads are good for graphical purposes to show a more intuitive (to me at least) picture of nested composition (substitution), but their in-line text notation is pretty gruesome compared to SMCs. And maybe the graphical advantage wouldn't even exist given the orthogonally drawn parameter wires of the para construction? It may just get highly cumbersome to represent visually. My motivation is I'm looking for a "best" way to encode and visualize multilevel substitution.

Kyle Wilkinson (Jul 31 2023 at 14:43):

Cole Comfort said:

Matteo Capucci (he/him) said:

On a different note, $\bf Para(\cal C)$ has a 2-dimensional structure I haven't seen in operads

There is a notion of 2-multicategories (essentially 2-operads), so I imagine that it would result in the same sort of thing.

Also coloured operads and monoidal categoires are not interchangable (although there is an adjunction between the category of coloured operads and monoidal categories). Monoidal categories are precisely the representable coloured operads; in a coloured operad you can't cut two circuits together along more than one wire at once

Is this the same as noting that any SMC gives rise to an operad, but not every operad arises from an SMC? I'm not sure what you mean by "cutting two circuits together along more than one wire at once." By the way, I suppose I should clarify that I am using the word operad to mean colored operad/multicategory.

Matteo Capucci (he/him) (Jul 31 2023 at 15:12):

Cole Comfort said:

Also coloured operads and monoidal categoires are not interchangeable (although there is an adjunction between the category of coloured operads and monoidal categories). Monoidal categories are precisely the representable coloured operads; in a coloured operad you can't cut two circuits together along more than one wire at once

Uhm right, I keep forgetting that