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Stream: deprecated: id my structure

Topic: Multicategories with restricted composition

Kevin Arlin (Aug 05 2023 at 21:49):

Imagine you have the object and multimorphism classes as for a multicategory, but the only composition operations you’re allowed are grafting unary morphisms onto domains and codomains of $n$-ary ones, to get another $n$-ary morphism. In particular the $n$-ary morphisms for $n\neq 1$ don’t interact at all with each other. Anybody happen to ever have met such a weird beast?

I suppose it falls apart into slightly more natural pieces, call them $n$-egories, where you have unary morphisms and $n$-ary morphisms for some fixed $n$ and you can compose $n$-ary with unary in the usual ways. Perhaps it’s best if the unary morphisms can’t even compose directly, but only act on the $n$-ary ones.

James Deikun (Aug 05 2023 at 21:55):

I guess you could model it as a multicategory that's free on a category; then the structure you describe would appear as the subset of morphisms shaped like elementary trees. Alternately, it's the data needed to construct a multicategory that's free on a category.

Kevin Arlin (Aug 05 2023 at 22:46):

Thanks, I hadn't thought about a free multicategory that has $n$-ary morphisms. I guess you mean to take wide cospans in your category as the multimorphisms? But since you mention elementary trees, are you composing them via tree grafting without actually composing in the underlying category?

Mike Shulman (Aug 05 2023 at 23:54):

To make math display here you have to use double dollar-signs: $$n$$ makes $n$.

James Deikun (Aug 06 2023 at 00:04):

What I mean by a "multicategory that's free on a category C" is in the sense of https://arxiv.org/abs/0907.2460 -- a free normalized $\mathbb{M}\mathsf{od}(\text{free monoid})$-monoid in $\bold{Set}\text{-}\mathbb{P}\mathsf{rof}$. This is generated by a bunch of generating $n$-morphisms with an action of $C$'s morphisms, like your above prescription. The generated multicategory can be seen as trees of generating $n$-morphisms with $C$-morphisms on the internal and external edges, identified up to the relation generated by having an edge act on an adjacent vertex and be replaced with $\mathsf{id}$.

James Deikun (Aug 06 2023 at 00:06):

(This quotienting means among other things that every multimorphism has at least one representing tree where all the $C$-morphisms along the edges are identities, so they aren't really needed for the representation, but they simplify the description of how to do the quotient.)

James Deikun (Aug 06 2023 at 00:19):

In the case where there's a single vertex in the tree, you get back the original generating data as unique representatives for each class of morphisms. In the case where there are zero vertices in the tree, you get a morphism of $C$ living on the edge, so I guess the morphisms of $C$ on the edges are not quite inessential--you need them for this one case.

Kevin Arlin (Aug 06 2023 at 00:38):

Ah, OK, very cool! The answer is pretty much always just “read the section of Mike’s relevant paper you didn’t read yet,” it seems.