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Cross reference ambidextrous adjunction.
Where it said
There is no relation to the notion of Frobenius monoidal functor,
I added
but there is a close relation to Frobenius monads.
Which highlights that this entry should be merged with ambidextrous adjunction, either way.
I realize now that “Frobenius functor” is (much) earlier terminology (though maybe less suggestive than “ambidextrous adjunction”?).
On the other hand, it seems that people who say “Frobenius functor” base their discussion on Morita 1965 who actually used yet another term, namely “strongly adjoint pair”.
Given this situation, I am inclined to merge everything into the entry ambidextrous adjunction. What do you think?
Re #2: as an aside, I’m often leery of claims that assert (emphatically!) that there is no relation between two concepts. (I want to ask the author: have you looked deeply into the matter? How can you be so sure?)
Something like this came up around here, where there was an assertion of “no relation” between Frobenius conditions (a la Frobenius monoids) and Frobenius reciprocity, but after investigation it emerged that there was a relatively close connection. It wouldn’t surprise me all that much if the same were true here.
The intent was to say that the terminology is independent: The definition of “Frobenius monoidal functors” is not “monoidal Frobenius functor”.
I have expanded the remark a fair bit (here)
also added the original reference for the terminology “Frobenius functor”:
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