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Stream: learning: questions

Topic: Frobenius-like identity

Nathanael Arkor (Jun 12 2021 at 10:38):

Is there a name for an object in a monoidal category with a multiplication $\mu \colon A \otimes A \to A$ and comultiplication $\delta \colon A \to A \otimes A$ satisfying the law
$(\mu \otimes 1) \circ (1 \otimes \delta) \circ (1 \otimes \mu) \circ (\delta \otimes 1) = \delta \circ \mu = (1 \otimes \mu) \circ (\delta \otimes 1) \circ (\mu \otimes 1) \circ (1 \otimes \delta)$? Possibly also satisfying (co)associativity. It looks a bit like a "double" Frobenius law, but I'm not sure where to look to find a reference.

Jason Erbele (Jun 21 2021 at 01:22):

@Nathanael Arkor, your "double" Frobenius law will hold for a special Frobenius algebra, but I'm not sure where you might look if you don't want to impose the usual Frobenius law.

Nathanael Arkor (Jun 21 2021 at 09:16):

Thanks! Yes, I had spotted that too, but I don't want to impose the usual Frobenius law (though I do actually want to impose the "special" part).

Amar Hadzihasanovic (Jun 21 2021 at 11:34):

That's interesting. I'll just remark that if your multiplication and comultiplication have a unit and counit, then “specialness” is a consequence of your equations (in fact, a consequence of either (1) = (2) or (2) = (3)).

Amar Hadzihasanovic (Jun 21 2021 at 11:35):

But I presume that your $\mu$ and $\delta$ are not unital, otherwise you would have said so.

Nathanael Arkor (Jun 21 2021 at 11:37):

Thanks – yes, I'm interested in cases where they're not (co)unital.

Amar Hadzihasanovic (Jun 21 2021 at 11:41):

Not trying to make this precise in any way, but the shape of the equations makes me think of some kind of “synchronisation via message passing”: (1) is “left system sends a message to right system, right system responds with a message to left system”; (3) is the same with right and left exchanged; and both are equivalent to (2) which is “left and right enter a joint system, then separate again”

Amar Hadzihasanovic (Jun 21 2021 at 15:19):

I guess if $\mu$ is the multiplication in a semigroup, not necessarily commutative, where each element is idempotent (e.g. a semilattice) and $\delta$ is the diagonal, then it satisfies your equations (and the specialness equation) but almost never the Frobenius equations.

Nathanael Arkor (Jun 21 2021 at 16:00):

That's a nice example.

Nathanael Arkor (Jun 21 2021 at 16:08):

I came across this equation in Kenney's The General Theory of Diads, but I haven't been able to find other references, despite it seeming like a (relatively) natural condition to impose.

Guillaume Boisseau (Jun 22 2021 at 22:54):

Amar Hadzihasanovic said:

I guess if $\mu$ is the multiplication in a semigroup, not necessarily commutative, where each element is idempotent (e.g. a semilattice) and $\delta$ is the diagonal, then it satisfies your equations (and the specialness equation) but almost never the Frobenius equations.

Aren't groupoids exactly special Frobenius algebras in Rel? https://arxiv.org/abs/1112.1284

Guillaume Boisseau (Jun 22 2021 at 22:55):

My bad you're taking $\delta$ to be the diagonal whereas this requires both (co)monoids to be the groupoid multiplication