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Stream: learning: questions

Topic: some free monoidal constructions

Paolo Perrone (Dec 21 2021 at 08:03):

Hi all. If I'm not mistaken, the 2-category of monoidal categories and lax monoidal functors, the 2-category of lax monoidal categories, and the 2-category of multicategories all have forgetful 2-functors to Cat.
Does any of those have a left adjoint?

Jules Hedges (Dec 21 2021 at 10:47):

Definitely the forgetful functor on 1-categories $\mathbf{MonCat} \to \mathbf{Cat}$ has a left adjoint that freely adds all monoidal products of objects in your category. It can be constructed eg. as a category of proof trees modulo commuting conversions, or a category of string diagrams modulo isotopy, in either case also modulo the equations in your original category. I'd guess that basically the same idea works in the 2-case, just with a lot more details...

Paolo Perrone (Dec 21 2021 at 10:50):

What are the morphisms of MonCat? I need lax monoidal functors, not strong or strict. (Oplax would work too.)

Jules Hedges (Dec 21 2021 at 10:55):

Oh, sorry, I misread that bit

Jules Hedges (Dec 21 2021 at 10:56):

Is it possible that there's a free functor $\mathbf{Cat} \to$ [monoidal categories, lax monoidal functors] whose image only hits strict monoidal functors?

Mike Shulman (Dec 21 2021 at 15:49):

The forgertful functor from multicategories to Cat definitely has a left adjoint, which just regards a category as a multicategory with only unary arrows.

Mike Shulman (Dec 21 2021 at 15:51):

I don't think the others do. I don't have a watertight argument, but consider that the forgetful functor $\rm MonCat_{strict} \to MonCat_{lax}$ does have a left adjoint, the [[lax morphism classifier]]. Thus, if the forgetful functor $\rm MonCat_{lax} \to Cat$ also had a left adjoint, then the forgetful functor $\rm MonCat_{strict} \to Cat$ would factor through it, meaning that every strictly-free monoidal category would be a lax morphism classifier. That seems impossible: e.g. a lax morphism classifier nearly always has nontrivial morphisms, whereas the strictly-free monoidal category on a discrete set never does.

Matteo Capucci (he/him) (Dec 21 2021 at 17:53):

@Mike Shulman your nLab link doesn't go anywhere, what's a lax morphism classifier? Actually, I might guess its definition: morphisms to it are equivalent to lax morphisms to some other fixed object. A better question is, how do you construct one?

Mike Shulman (Dec 21 2021 at 17:54):

Huh, sorry, I thought we had that page, but I guess not. Strict morphisms from the lax morphism classifier of A are equivalent to lax morphisms from A. Probably the clearest reference for its construction is Steve Lack's Codescent objects and coherence.

Paolo Perrone (Dec 21 2021 at 21:40):

Mike Shulman said:

I don't think the others do. I don't have a watertight argument, but consider that the forgetful functor $\rm MonCat_{strict} \to MonCat_{lax}$ does have a left adjoint, the [[lax morphism classifier]]. Thus, if the forgetful functor $\rm MonCat_{lax} \to Cat$ also had a left adjoint, then the forgetful functor $\rm MonCat_{strict} \to Cat$ would factor through it, meaning that every strictly-free monoidal category would be a lax morphism classifier. That seems impossible: e.g. a lax morphism classifier nearly always has nontrivial morphisms, whereas the strictly-free monoidal category on a discrete set never does.

I see. Very good argument. Thank you!