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Stream: learning: questions

Topic: Is taking companions a symmetric monoidal pseudofunctor?

John Baez (Sep 01 2025 at 16:12):

In @Mike Shulman's work on constructing a symmetric monoidal bicategory $\cal{L}(\mathbb{D})$ from a symmetric monoidal double category $\mathbb{D}$, I would expect to see a remark saying that taking companions gives a symmetric monoidal pseudofunctor from

• the category where morphisms are tight morphisms of $\mathbb{D}$

to

• the bicategory where morphisms are loose morphisms of $\mathbb{D}$, namely $\cal{L}(\mathbb{D})$.

But I'm not seeing that. So I'm wondering:

1) Has someone stated this somewhere?
2) Is it even true?

John Baez (Sep 01 2025 at 16:19):

In the case where $\mathbb{D}$ is the double category of cospans in some category with finite colimits it seems "obvious", and that's the only case I really need, but I'd like it to be true more generally, as above.

Mike Shulman (Sep 01 2025 at 16:45):

I don't think I/we wrote this down. But I think you should be able to deduce it from the functoriality results we did write down, if not "easily" then at least without further delving into the definition of symmetric monoidal bicategory or pseudofunctor. If $T(\mathbb{D})$ denotes the category of tight morphisms, then the double category $Q(T(\mathbb{D}))$ of "quintets" in $T(\mathbb{D})$ should be a symmetric monoidal double category with companions, and taking companions should be a symmetric monoidal double functor $Q(T(\mathbb{D})) \to \mathbb{D}$ (most of whose constraints are identities). And $L(Q(T(\mathbb{D}))) = T(\mathbb{D})$, so this symmetric monoidal double functor should induce, by the existing functoriality statement, a symmetric monoidal pseudofunctor $T(\mathbb{D}) \to L(\mathbb{D})$.

John Baez (Sep 01 2025 at 16:56):

Thanks! That's a clever way to avoid certain kinds of mucking around.

I'm still hoping @Nathanael Arkor or someone has already proved this, since this fact was supposed to be a quick step in a longer argument.

Nathanael Arkor (Sep 01 2025 at 17:22):

I don't know anywhere this is proven. The fact that taking companions gives you a pseudofunctor is well known, so it only remains to verify symmetric monoidality, which I expect would be straightforward, because the monoidal structure on $\mathcal L(\mathbb D)$ is defined in terms of that on $\mathbb D$. This is less slick than Mike's suggestion, although I suppose there's some manual verification to do in either case.

John Baez (Sep 01 2025 at 20:20):

Thanks. Is that well known fact that taking companions gives you a pseudofunctor written down anywhere?

Mike Shulman (Sep 01 2025 at 20:25):

It's asserted without explicit proof in Proposition 4.16 of Framed bicategories and monoidal fibrations.

Evan Patterson (Sep 01 2025 at 21:33):

A version of this result (with a fairly complete proof invoking a series of technical lemmas) is Theorem 2.15 in my paper Transposing cartesian and other structure in double categories.

John Baez (Sep 01 2025 at 22:22):

Thanks! I'm trying to understand this....