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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?