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Stream: theory: category theory

Topic: Symmetric monoidal bicategory of a SM Double category

David Jaz Myers (Jan 31 2026 at 18:09):

In his paper Constructing symmetric monoidal bicategories, Shulman shows that if a symmetric monoidal double category has all companions and conjoints, then its horizontal bicategory is also symmetric monoidal.

It seems clear to me that it suffices for the double category to only have all companions and conjoints for isomorphisms (that is, it suffices for it to be isofibrant, rather than fibrant). This is simply because we only ever use companions and conjoints of the structural isomorphisms of the symmetric monoidal structure in the proof.

Now, if a vertical isomorphism $a$ and its inverse $b$ have companions, then the companion to $b$ will be a conjoint for $a$. Therefore, it seems that in the key Theorem 4.6 of ibid., it suffices to assume that all vertical isomorphisms have companions, or in other words that the double category is left isofibrant (it's source map is an isofibration).

Is this reasoning correct, or have I missed something?

John Baez (Feb 01 2026 at 05:08):

It might be worth pinging @Mike Shulman.

Nathanael Arkor (Feb 01 2026 at 09:53):

The non-symmetric version also follows from the work of Garner and Gurski, who merely ask for $\mathbb D_1 \to \mathbb D_0 \times \mathbb D_0$ to be an isofibration.

Nathanael Arkor (Feb 01 2026 at 09:54):

(And for symmetry one does not need stronger assumptions.)

Nathanael Arkor (Feb 01 2026 at 09:59):

The reasoning about companions of tight isomorphisms is essentially Theorem 1.5 of Grandis–Paré's Adjoint for double categories

Nathanael Arkor (Feb 01 2026 at 17:02):

(I don't see that asking for the source functor to be an isofibration is the same as asking for every tight isomorphism to have a companion, though.)

Mike Shulman (Feb 02 2026 at 19:34):

Yes, see Remark 3.22 in my paper.

David Jaz Myers (Feb 03 2026 at 09:18):

Mike Shulman said:

Yes, see Remark 3.22 in my paper.

Ah great, sorry I missed that!

Thanks all!