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Patrick Nicodemus (May 13 2026 at 16:55):Hi all,
I am aware that Tom Leinster's notion of "generalized operad" can be used to characterize categories as algebras of a generalized operad in Graph.
I am interested in whether "generalized multicategories" or a related notion are documented for characterizing the following standard concepts as algebras of a generalized operad / multicategory.
I am currently working through these characterizations on my own and I have characterized functors as generalized multicategory algebras to my own satisfaction. I would appreciate pointers to existing literature which model these "compound" structures involving multiple sorts as models of a certain "algebraic theory" which shares some nice relationship with operads - it need not be Leinster's formulation.
El Mehdi Cherradi (May 16 2026 at 09:35):Hello,
https://arxiv.org/abs/0907.2460 is relevant here in case you have not looked at it.
Patrick Nicodemus (May 20 2026 at 17:13):Hi all.
As I said above, it is well known that categories are algebras for a generalized multicategory in , which Leinster calls the "suspension" of the operad for monoids. (It took me a long time to realize that what Leinster calls the "suspension" is simply the horizontal categorification of the theory.)
I have come to the conclusion that
But I was unable to cast adjunctions in this light. I don't claim it is impossible, but if we recall that generalized multicategories over are exactly the virtual double categories, and that algebras for these g.m.c's are essentially functors on the V.D.C. valued in the V.D.C. of sets and spans, then it becomes a bit more clear why it is hard to express adjunctions as a VDC - the map involves restriction of the Hom functor, which means that we would really like to express this theory as a virtual equipment.
This is nice but it seems to involve a departure from the existing theory of operads and generalized multicategories. If our goal is to find common frameworks for talking about categorical coherence in different situations, we know that "contractible operads" are a useful tool. The paper linked above by Cruttwell and Shulman contains on page 6 a table of many different kinds of generalized multicategories in their framework, but virtual equipments are not among them. Is the conclusion here that the theory of an adjunction is not well expressed an "operadic" theory because it involves restriction? Is there any well known class of algebraic structures which include certain generalized multicategories and also certain virtual equipments?
Patrick Nicodemus (May 20 2026 at 21:58):I think I have worked something out that unifies these notions.
Patrick Nicodemus (May 28 2026 at 13:02):I have drafted a short paper which observes that, just as virtual double categories are a special case of generalized multicategories, virtual double categories with all restrictions are a special case of some kind of "generalized multicategory with restrictions", which I define in the paper. I also define the notion of an algebra for a generalized multicategory with restriction, which generalizes functors from a virtual equipment into Set preserving Cartesian 2-cells.
I would be interested in comments on the mathematical content - I understand it is not especially polished, but please say something if any proof is hard to follow or unclear.
@John Baez Would you be interested in skimming this and suggesting a next direction? I don't think the diagrams are especially intuitive or clarifying, but being able to extend the theory of operads with "reindexing" strikes me as plausibly powerful.