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Stream: learning: questions

Topic: parametrized multicategories

Jonas Frey (Aug 06 2026 at 10:19):

Let $C$ be a multicategory. By a $C$-parametrized multicategory $D$ I mean a structure which has a set/class $D_0$ of morphisms, and multi-arrows which may have multiple inputs from $C$ as well as $D$, and a unique output in $D$.

Has this notion been studied?

It would generalize [[multiactegories]] by allowing multiple $D$-inputs rather than a unique one for the morphisms.

Noam Zeilberger (Aug 06 2026 at 12:19):

Would @Mike Shulman's LNL polycategories (or at least their restriction to LNL multicategories) be an example of the sort of structure you have in mind?

Jonas Frey (Aug 10 2026 at 14:44):

Yes, thanks! My phrasing didn't impose non-linear structural rules on the parametrizing multicategory, but apart from that it seems to be the same as an LNL multicategory.

Jonas Frey (Aug 10 2026 at 14:50):

The poly-generalization that Shulman considers is interesting, but I haven't come across any examples naturally, contrary to the multi case.

Jonas Frey (Aug 10 2026 at 14:51):

But of course that might be because of my prejudiced outlook ...

Mike Shulman (Aug 15 2026 at 19:03):

Also worth noting that a pair consisting of C and a C-parametrized multicategory D is equivalent to a multicategory equipped with a functor to the multicategory freely generated by a monoid and an algebra over it.

Jonas Frey (Aug 15 2026 at 20:06):

Is C the domain of the functor? And what do you mean by algebra over a monoid ?

Mike Shulman (Aug 15 2026 at 20:21):

Let me just be explicit: by "the multicategory freely generated by a monoid and an algebra over it" I mean the multicategory with two objects $c,d$, one morphism $(c,\dots,c) \to c$ of every arity (making $c$ a monoid), and one morphism $(c,\dots,c,d,\dots,d) \to d$ of every pair of arities. Then $C$ is the fiber over $c$.

Jonas Frey (Aug 15 2026 at 20:28):

Thanks, that makes sense!