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Stream: theory: category theory

Topic: pseudo multicategories

Mike Shulman (Aug 08 2024 at 18:07):

Is there anywhere in the literature containing a precise definition and study of what one might call "pseudo multicategories" -- that is, multicategories weakly enriched over Cat, in the same way that bicategories are categories weakly enriched over Cat? Writing down such a definition is easy, of course, but I can't think of anywhere citable that I've seen it done.

Evan Patterson (Aug 08 2024 at 18:17):

Have you seen this paper? https://arxiv.org/abs/1603.02146

It's not quite what you're looking for inasmuch as it treats 2-multicategories in the strict rather than pseudo sense, but it does define pseudo functors and pseudo natural transformations for 2-multicategories.

Mike Shulman (Aug 08 2024 at 18:19):

That's certainly something! Thanks.

Nathanael Arkor (Aug 08 2024 at 18:21):

There's a definition of "bi-multicategory" in Definition 4.2.29 of @Philip Saville's thesis Cartesian closed bicategories: type theory and coherence.

Mike Shulman (Aug 08 2024 at 18:22):

Ah, even better!

Nathanael Arkor (Aug 08 2024 at 18:23):

(The appropriate terminology for such a structure is slightly subtle. In particular, for consistency "pseudo multicategory" ought to refer to a generalisation of a pseudo category, rather than a generalisation of a bicategory. Furthermore, "multibicategory" is already used in the literature to refer to what might alternatively be called a "virtual bicategory".)

Mike Shulman (Aug 08 2024 at 18:24):

Yeah. Probably "weak 2-multicategory" is the least ambiguous.

Mike Shulman (Aug 08 2024 at 23:27):

Although "virtual monoidal bicategory" would also be systematic.

Noah Chrein (Aug 16 2024 at 16:31):

Been working on a notion of quasi-multicategory, i.e. a simplicial generalization of multi (categories). I wonder if it would be of use.

Kevin Carlson (Aug 16 2024 at 21:09):

How would such a thing relate to $\infty$-operads in the style of Lurie or that of Cisinski-Moerdijk?

Noah Chrein (Aug 19 2024 at 15:57):

I imagine the "one"-object (something like contractible) $\infty$-multicategories would be the $\infty$-operads.

Kevin Carlson (Aug 19 2024 at 16:49):

I think those authors usually use “operad” to imply “colored”, ie multi-object, though.

Noah Chrein (Aug 22 2024 at 16:55):

I see. Then it seems the dendroidal sets of https://ncatlab.org/nlab/show/(infinity,1)-operad#in_terms_of_dendroidal_sets
would coorespond to $\infty-T$-multicategories with T = free monoid on sets.

Noah Chrein (Aug 22 2024 at 16:56):

which is to say $\infty-fc$-multicategories would be like $\infty$-virtual double categories etc.

(here the "$\infty$" is in the category direction, not the virtual direction)

Kevin Carlson (Aug 22 2024 at 18:29):

Yeah, that sounds right.