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Stream: learning: questions

Topic: Cartesian Multicategories and Operads

Ben Logsdon (Nov 09 2023 at 17:50):

The goal of this question is to collect basic information about (the categories of) Cartesian multicategories and Cartesian operads. Hopefully the result will be useful to others as well.

I mean "Cartesian multicategory" in (I think) the standard sense (see "Higher Operads, Higher Categories" by Leinster). By "Cartesian operad", I just mean a Cartesian multicategory with one object.
Denote the categories of Cartesian multicategories and Cartesian operads by $\mathfrak{CMult}$ and $\mathfrak{COp}$ , respectively.

I'm particularly interested in the following questions; answers to any or all are appreciated.

Are there any other names for "Cartesian multicategory" and "Cartesian operad" that are commonly used in the literature? Is "Cartesian operad" a commonly used name?
What categorical structure/properties are (and are not) possessed by $\mathfrak{CMult}$ and $\mathfrak{COp}$ ? I'm interested in anything, from the more plain (limits, colimits, etc.) to the exotic.
What are some references that discuss the basic properties of Cartesian multicategories, Cartesian operads, $\mathfrak{CMult}$ , and $\mathfrak{COp}$ ? (Including references for the properties mentioned for the above question)

A note on motivation: My motivation from this question stems from categorical logic. Cartesian multicategories can serve as semantics for (multi-sorted) algebraic theories (or perhaps are algebraic theories, depending on perspective). Likewise, Cartesian operads are (semantics for) Lawvere theories (single-sorted algebraic theories). From this viewpoint, structure on $\mathfrak{CMult}$ and $\mathfrak{COp}$ corresponds to operations that one can perform on algebraic/Lawvere theories.

Kevin Arlin (Nov 09 2023 at 18:03):

I don't know much of anything about references for these things other than Shulman's draft book, but multicategories, symmetric multicategories, and cartesian multicategories are all locally finitely presentable for basically the same reason as categories: you have some objects, some (sets of) arrows, some identity assignment functions, and some (sets of) composable pairs, forming a presheaf category in which your category variant is fully and accessibly embedded. I know that multicategories are not cartesian closed and symmetric multicategories are; I'm not quite sure about cartesian multicategories but my first guess would be that they are cartesian closed.

Kevin Arlin (Nov 09 2023 at 18:04):

It's notable that in being lfp these categories are much better behaved than closely related categories like that of (symmetric/cartesian) monoidal categories and lax monoidal functors.

Mike Shulman (Nov 09 2023 at 18:16):

The phrases "cartesian multicategory" and "cartesian operad" don't appear in Leinster's book. Do you mean the one I wrote about and that's on the nlab [[cartesian multicategory]]? These are also called "abstract clones" so you can sometimes find more information by searching for that.

Mike Shulman (Nov 09 2023 at 18:19):

Cartesian multicategories and operads also appear briefly (though not under that name) as Example 4.17 in my paper A unified framework for generalized multicategories with Geoff Cruttwell.

Ben Logsdon (Nov 09 2023 at 18:29):

Mike Shulman said:

The phrases "cartesian multicategory" and "cartesian operad" don't appear in Leinster's book. Do you mean the one I wrote about and that's on the nlab [[cartesian multicategory]]? These are also called "abstract clones" so you can sometimes find more information by searching for that.

Oh yes, you're right. I was thinking of the nLab's definition.

Claudio Pisani (Nov 18 2023 at 20:31):

Kevin Arlin said:

I know that multicategories are not cartesian closed and symmetric multicategories are;

I don't think that the category $\rm sMlt$ of symmetric multicategories is cartesian closed.
In particular, the unary ones $\,\,\cal C^-$ are not exponentiable.
Consider in particular $\,\,1^- \in \rm sMlt$ , the symmetric multicategory with just one object and one arrow (the identity). For any $\,\, M\in \rm sMlt$ , $\,\,1^-\times M = M_-$ , the underlying unary multicategory, and the functor $(-)_-$ does not preserve pushouts (and so it cannot have a right adjoint $(-)^{1^-}$ ).

Claudio Pisani (Nov 18 2023 at 20:34):

For instance, consider $\,\, M\in \rm sMlt$ with three objects, $A,B,C$ , and generated by a single arrow $A,B\to C$ and $\,\, N\in \rm sMlt$ with an object $D$ and a nullary arrow $\,\,\to D$ . Let $P$ be the pushout of $M$ and $N$ along the objects $A$ and $D$ .
Then $P$ has a non-identity arrow $B\to C$ (the composite of the nullary one and the ternary one) so that $\,\,1^-\times P$ is not discrete, while both $\,\,1^-\times M$ and $\,\,1^-\times N$ are discrete.

Claudio Pisani (Nov 18 2023 at 20:42):

May be you refer to the Boardman-Vogt monoidal closed structure on $\rm sMlt$ , but it is not the cartesian one.

Claudio Pisani (Nov 18 2023 at 20:59):

As discussed here, the exponentiable multicategories are the promonoidal ones, in the sense of B. Day, E. Panchadcharam, R. Street (2005), and this should be true also for the symmetric ones.

Claudio Pisani (Nov 18 2023 at 22:18):

These promonoidal=exponentiable multicategories are those $M$ such that for any arrow in $M$ (say, $f : A,B,C \to D$ ) and any "factorization of its shape" (say as two binary arrows)
1) there is an actual factorization $f = g\circ h$ following that shape (with, say, $h:A,B \to E$ and $g:E,C \to D$ ) and
2) any two such factorizations are "equivalent", where the equivalence is generated by $g'\circ th \simeq g't\circ h$ , for any unary arrow $t:E\to E'$ and $g': E',C \to D$ , the factorization

Claudio Pisani (Nov 18 2023 at 22:37):

It is interesting to note that if one considers the "free prop" $PM$ generated by $M$ , that is the category whose objects are families of objects in $M$ and whose arrows are families of arrows in $M$ , with its obvious indexing functor to $F:PM \to \rm Set_f$ , then $M$ is promonoidal=exponentiable if and only if $F$ itself is exponentiable as a category over $\rm Set_f$ .

Claudio Pisani (Nov 18 2023 at 22:42):

Indeed, it is easy to see that the above condition become the usual factorization lifting condition of Conduché functors = exponentiable functors.

Claudio Pisani (Nov 18 2023 at 23:08):

Furthermore, since symmetric monoidal categories arise (in their unbiased form) as those $M\in \rm sMlt$ for which the above $F:PM \to \rm Set_f$ is an opfibration, and since (op)fibrations are Conduché functors, symmetric monoidal categories are exponentiable in $\rm sMlt$ .

Kevin Arlin (Nov 20 2023 at 06:28):

Thanks for the comments. Yes, I misspoke about the cartesianness of the closure. This is a case where I think the closed structure is really primary, which might explain my misdescribing its left adjoint.