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Stream: theory: category theory

Topic: Fibrations of multicategories

Evan Patterson (Aug 26 2023 at 21:57):

I am trying to find and understand a notion of [[fibration of multicategories]] that will work for my application. From what I can gather from the literature, there are at least two possible notions of fibration.

I think I want to have cartesian liftings against unary morphisms $f: x \to y$ and have the universal property of a cartesian multimorphism apply to factorizations of the form $(x_1,\dots,x_n) \xrightarrow{g} x \xrightarrow{f} y$. In the paper Fibrations for abstract multicategories (which is perhaps more like an extended abstract), Hermida says that "fibrations in Multicat" have "cartesian liftings of linear morphisms only". I take this to mean that if I were to unwind the scary-looking definition of a [[fibration in a 2-category]] for the 2-category of multicategories, multifunctors, and natural transformation I would get the notion I am after. Am I on the right track here?

Mike Shulman (Aug 28 2023 at 16:25):

That's also my best guess as to what he meant. And it might even be true.

Evan Patterson (Aug 28 2023 at 16:59):

Thanks! Maybe I'll try to sort this out.

Dylan Braithwaite (Aug 28 2023 at 17:43):

@Nicolas Blanco may have thought about this during his work on bifibrations of polycategories? Though taking a quick glance at the definition of pull-fibration in his thesis, I can't tell if this is equivalent to the definition you sketched (when specialised to multicategories).

Mike Shulman (Aug 28 2023 at 17:46):

That's more general, it doesn't require the morphism to be unary.

Dylan Braithwaite (Aug 28 2023 at 17:52):

That's what I thought. I was just thrown a bit by the fact that you get to choose all-but-one of the domain objects in the definition of lifting.

I can see that there doesn't seem to be an obvious way to define such a thing from a fibration with unary lifts though

Mike Shulman (Aug 28 2023 at 17:54):

Yes, the idea is that the more general kind of fibration also incorporates operations like internal-homs.