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Stream: learning: questions

Topic: Functors of Fibrant Double Categories via Proarrow Equipment

Joshua Meyers (Mar 27 2024 at 14:09):

A (weak) fibrant double category $\mathbb{D}$ induces a structure similar to a proarrow equipment but slightly weaker consisting of the following data:

• A bicategory $\mathcal{D}$ which is the horizontal bicategory of $\mathbb{D}$
• A category $\mathbb{D}_0$ which is the vertical category of $\mathbb{D}$
• An identity-on-objects pseudofunctor $\overline{(-)}:\mathbb{D}_0\to\mathcal{D}$ sending each vertical arrow to its companion, such that every morphism in the image is a left adjoint in $\mathcal{D}$

(this is differs from a proarrow equipment in that $\mathbb{D}$ is a 1-category)

In @Mike Shulman's paper "Framed Bicategories and Monoidal Fibrations", he cites Dominic Verity who defined morphisms of these equipments essentially by making them into double categories and then using double functors --- so it would be tautologous to get morphisms of fibrant double categories from morphisms of eqiupments. However, I am not so sure.

Conjecture:

Let $\mathbb{D}$ and $\mathbb{E}$ be fibrant double categories. Then a strong double functor aka strong framed functor $F:\mathbb{D}\to\mathbb{E}$ is equivalent to the following data:

• A pseudo functor $F:\mathcal{D}\to\mathcal{E}$
• A functor $F:\mathbb{D}_0\to\mathbb{E}_0$ agreeing with the pseudo functor $F$ on objects (this allows us to use the same name $F$ without ambiguity)
• A pseudo natural transformation $\overline{(-)}\circ F\cong F\circ\overline{(-)}:\mathbb{D}_0\to\mathcal{E}$ sending objects to identities.

Is there any known result like this?

Joshua Meyers (Mar 27 2024 at 18:42):

Also ... a similar notion of morphism of proarrow equipment occurs in Proarrows II by R. J. Wood, starting at the bottom of p. 163

Mike Shulman (Mar 27 2024 at 21:17):

Yes, I think that's right, although I don't know where it appears in the literature.

Mike Shulman (Mar 27 2024 at 21:17):

What is it you aren't so sure about?

Joshua Meyers (Mar 27 2024 at 23:33):

Well I'm not sure of anything in math until I see a proof. And the structures involved are complicated enough that I can't write a proof quickly -- it would take me a while. Moreover, I would have assumed that a result so basic would be in the literature somewhere and not only is it not, but you remark in "Framed Bicategories and Monoidal Fibrations" that Verity resorts to constructing double categories and just using morphisms of those --- which seems to imply that there is something lacking in a naive notion of equipment morphism.

Joshua Meyers (Mar 28 2024 at 02:16):

More specifically, I'm not sure if there's some kind of coherence condition which is missing / too strong.

Mike Shulman (Mar 28 2024 at 14:20):

I think it's more that without the double-category perspective one is less likely to guess the "right" notion of equipment morphism, not that it's especially difficult to write down once you have it.

Mike Shulman (Mar 28 2024 at 14:21):

The notion of "equipment 2-cell" is even less obvious without the double-category perspective.

Nathanael Arkor (Apr 18 2024 at 10:56):

The relationship between morphisms of proarrow equipments and morphisms of fibrant double categories is discussed in section 4.1.3 of Lawler's Fibrations of predicates and bicategories of relations.