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Stream: theory: category theory

Topic: higher operads

Reuben Stern (they/them) (Oct 07 2022 at 14:50):

Is there a developed notion of a "double operad", where imposing some completeness condition would give a "2-operad" (by which I mean a (2,2)-operad)? My motivation comes from duoidal categories. There should be some object that acts like an "$\mathbb{E}_{1+1}$" operad, which is a "lax Boardman-Vogt tensor product" of the associative operad with itself, algebras over which in $\mathsf{Cat}$ give duoidal categories. Ultimately I am interested in the $(\infty,2)$ version

Nathanael Arkor (Oct 07 2022 at 14:56):

By "double" do you mean double categorical, or something else? And if the former, what kind of cocompleteness condition do you have in mind?

Reuben Stern (they/them) (Oct 07 2022 at 14:59):

Yes, I mean double categorical. And for completeness, I'm really not sure -- something resembling completeness as in a complete Segal space perhaps (in the same way that you get an $(\infty,2)$-category from a double $\infty$-category by imposing a completeness condition)

Nathanael Arkor (Oct 07 2022 at 15:06):

I think a low-dimensional definition of double operad would not have any notion of completeness in this sense: I would expect every double operad to induce a 2-operad simply by forgetting one of the dimensions. In any case, the only definition of double operad I have encountered is in the paper of Cheng–Gurski–Riehl Multivariable adjunctions and mates, who give one possible definition of "double multicategory" (after which you can restrict to a single object for double operads). I suspect there are more general definitions that could warrant the name "double multicategory", though, so it depends exactly what structures you're interested in.

Reuben Stern (they/them) (Oct 07 2022 at 16:01):

Thanks for that reference, I'll take a look through that. When you say a double operad should induce a 2-operad by forgetting one of the dimensions, I think you're referring to a different idea of 2-operad than what I was asking for (a (2,1)-operad rather than a (2,2)-operad). With the definition of a double multicategory as a category object in multicategories, maybe what I'm looking for is no more than a category object in $\infty$-operads, perhaps then imposing a completeness condition like "levelwise completeness" where by "levelwise" i mean "for each finite set", since by straightening I am thinking of an $\infty$-operad as a functor $\mathsf{Fin}_* \to \mathsf{Cat}_\infty$.

Nathanael Arkor (Oct 07 2022 at 16:08):

When you say a double operad should induce a 2-operad by forgetting one of the dimensions, I think you're referring to a different idea of 2-operad than what I was asking for (a (2,1)-operad rather than a (2,2)-operad).

I mean that a (coloured) double operad should contain objects, a horizontal component, a vertical component, and 2-cells. One then obtains a 2-operad by forgetting the vertical component. (Similarly to how we obtain a 2-category from every double category by forgetting the vertical arrows.)

Reuben Stern (they/them) (Oct 07 2022 at 16:44):

Got it, I was confused.

Mike Shulman (Oct 07 2022 at 17:18):

Nathanael Arkor said:

I suspect there are more general definitions that could warrant the name "double multicategory", though, so it depends exactly what structures you're interested in.

For this reason, I usually refer to a category object in multicategories as a "multi double category" instead.