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Quentin Schroeder (Apr 11 2025 at 12:56):On the nlab entry for lax idempotent 2-monads (KZ-monads), there is a claim that for "some" pseudomonads (on Cat let's say) T wrt which we can define T-multicategories, we get a lax idempotent monad on the T-multicategories for which the pseudo algebras are the algebras of the original monad. The motivating example is that of multicategories and monoidal categories, where "being represetanble" is a property of a multicategory.
My questions are:
Is there a reference for the claim about T-multicategories?
If so what are the conditions for this to happen (I have some guesses but nothing concrete)?
James Deikun (Apr 11 2025 at 13:47):The relevant, last paragraph of the Examples section in [[KZ-monads]] was added by @Mike Shulman (revision 6, way back in 2010) ...
James Deikun (Apr 11 2025 at 14:05):My best guess as to a reference would be Remark 9.16 in Cruttwell and Shulman which doesn't actually exhibit any conditions either but promises them in a paper "Representability of generalized multicategories" by the same authors that I think never actually got published.
Quentin Schroeder (Apr 11 2025 at 14:41):Thanks a lot! I'll take a look! I'd love to know if this is still something in the works.
Nathanael Arkor (Apr 11 2025 at 14:47):The reference is Hermida's From coherent structures to universal properties, which is at the generality of Burroni's -categories for a cartesian monad on a nice category .
Nathanael Arkor (Apr 11 2025 at 14:50):@Miloslav Štěpán's recent thesis Lax structures in 2-category theory also contains relevant content in section 6.2.
James Deikun (Apr 11 2025 at 15:25):Hermida is definitely a more useful reference, even if it probably wasn't the one the author had in mind.
Mike Shulman (Apr 11 2025 at 19:17):Yes, I think Hermida's paper is the best extant reference. Sadly, that promised paper of Geoff's and mine never appeared, although I suppose it still might one day.
John Baez (Apr 11 2025 at 21:39):"Extant reference" - sounds like something a classicist would say about the commentary tradition on Aristotle.