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Stream: learning: questions

Topic: Frobenius pseudomonoids in a bicategory of cospans

John Baez (Aug 31 2025 at 10:18):

Let $\mathsf{A}$ be a category with finite colimits and $\mathbf{Csp}(\mathsf{A})$ the bicategory of cospans in $\mathsf{A}$.

Then every object $a$ of $\mathbf{Csp}(\mathsf{A})$ is a [[Frobenius pseudomonoid]] where the multiplication is the cospan

$a + a \to a$

coming from the codiagonal morphism in $\mathsf{A}$ and the unit is the cospan

$0 \to a$

coming from the unique morphism from the initial object of $\mathsf{A}$, and the pairing is the obvious cospan

$a + a \to a \to 0$

where $a + a \to a$ is the multiplication, and $a \to 0$ comes from turning around the cospan $0 \to a$.

This is fairly easy to show, but has anyone published a proof yet?

Furthermore this Frobenius pseudomonoid should be a categorified version of a special Frobenius monoid, but has anyone defined special Frobenius monoids yet, and proved this?

Cole Comfort (Sep 04 2025 at 01:54):

This is discussed for the 1-category of cospans internal to the skeleton of finite sets in lack's paper "Composing Props:"

http://www.tac.mta.ca/tac/volumes/13/9/13-09.pdf

It generalizes to finitely cocomplete categories relatively straightforwardly, as you suggest, so I guess that part is just folklore.

For the bicategorical generalization of a special Frobenius monoid, I suppose that there is no canonical choice of definition. I am not aware if the specific result you want is published anywhere.

You may want to ask for something like the locality axioms introduced here:

https://youtu.be/rZp1dPJxG3I

It would also be natural to ask that the multiplication of the monoid is left adjoint to the comultiplication of the comonoid, where the counit for the adjunction is an isomorphism, in addition to the "locality" axioms. Maybe you also want to ask that the unit of the monoid is left adjoint to the counit of the comonoid.

John Baez (Sep 04 2025 at 08:32):

It would also be natural to ask that the multiplication of the monoid is left adjoint to the comultiplication of the comonoid, where the counit for the adjunction is an isomorphism...

I guess that is your categorification of the 'special' law?

Cole Comfort (Sep 04 2025 at 09:46):

John Baez said:

It would also be natural to ask that the multiplication of the monoid is left adjoint to the comultiplication of the comonoid, where the counit for the adjunction is an isomorphism...

I guess that is your categorification of the 'special' law?

Yeah that's the idea I had. But I don't know exactly what the "right" categorification would be in this context. I am pretty sure that in copspan this multiplication is left adjoint to the comultiplication, so it seems natural to include them in the coherence data by asking that the "special" law is witnessed by the counit of this adjunction being an isomorphism. There are possibly more induced natural transformations and coherences which you may want to include in the definition of the categorification.

Cole Comfort (Sep 04 2025 at 09:55):

Lack proves that the walking special commutative frobenius monoid is symmetric monoidally equivalent to cospans of finite sets under the coproduct by showing that the pushout is witnessed by the distributive law of props between the walking commutative monoid and the walking cocommutative comonoid.

So maybe the "right" definition in this context would be to find the right categorification of this distributive law, or equivalently the right categorification of this strict factorization system. Then translate the coherences into the definition. Just a suggestion though, I would be interested if you find the solution!

John Baez (Sep 04 2025 at 10:51):

To me the "right" categorification of the special law is whatever law or laws that hold in the symmetric monoidal bicategory of cospans (in a category with finite limits) that are not implied by the "symmetric Frobenius pseudomonoid" laws.

(Of course there might be some wholly new laws, which would be thrilling, but there must at least be some that reduce to the special law when we decategorify and work with the mere symmetric monoidal category of spans.)