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Stream: theory: category theory

Topic: Pseudomonoidal fibrant double categories

Hugo Paquet (Nov 21 2022 at 10:35):

Often, it is helpful to view a bicategory $\mathcal{B}$ as the horizontal part of a fibrant double category $\mathbb{B}$. The reason is that in a double category we can talk about isomorphisms between objects, using the vertical structure, whereas in a bicategory we can only talk about equivalence.

One useful application is that we can transport any monoidal structure from $\mathbb{B}$ to $\mathcal{B}$. The idea is that the structural
isomorphisms in $\mathcal{B}$ are obtained as "companions" of the structural isomorphisms in $\mathbb{B}$. (This works for other kinds of structure, see Shulman, Hansen and Shulman, Gambino, Garner, and Vasilakopoulou.)

It is usually difficult to define a monoidal bicategory because we also need to give a number of structural modifications with intimidating coherence axioms. But here, these are induced in a canonical way, using that two companions of the same vertical 1-cell are canonically isomorphic. The axioms hold because the canonical isomorphisms are closed under composition, tensor, etc.

Is there a way to do this when the monoidal structure in $\mathbb{B}$ is only pseudo? I mean, if axioms only hold up to structural modifications (between vertical transformations), that satisfy the same axioms as for a monoidal bicategory.
This is sometimes still easier to do than to verify everything in $\mathcal{B}$ directly. The argument above doesn't seem to apply, but it still feels like something should be possible.

Mike Shulman (Nov 21 2022 at 16:27):

Yes, it should work. The pseudofunctor from the tight 2-category to the loose bicategory (I prefer those words to "vertical" and "horizontal" now) is fully faithful, so asserting all the structure in the tight case is basically the same as asserting it in the loose case and giving tight representatives of the morphisms.

Mike Shulman (Nov 21 2022 at 16:28):

I don't know if anyone's written it down, though.