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I started an entry at nucleus of a profunctor. Much more could be said, including plenty of examples. I don’t know what people think about splitting examples into those where the profunctor is Hom from the rest.
Do we not already have something on the ’center of an adjunction’, perhaps under a different name?
We have Legendre transfomation already. Is there a standard way to take the Legendre-Fenchel transform as a generalization?
I have a conference talk on duality to give in September, hence my looking at the above.
If at Isbell duality, we have
Isbell duality is a template for many other space/algebra-dualities in mathematics,
and this is a special case of the duality formed by this nucleus construction, is there a reason that choosing Hom as the profunctor leads to space/algebra dualities, or should we expect such dualities to arise in non-Hom cases?
I suppose it would be worthwhile to look at more general dualities, but at least one can say of course that Hom is in some way the canonical/archetypical/tautological example and hence somehow special or somehow more fundamental than other examples will be.
But the truth-valued Galois correspondence ones don’t come from Hom, as listed here.
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