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Wiki has interesting chapter https://en.wikipedia.org/wiki/Adjoint_functors#Solutions_to_optimization_problems that adjoint functors can be used for optimization, I guess more in the sense of finding optimal objects, structures. Is this original idea whose first exposition is in the wiki article or maybe there are available some references and elaborations of this idea? It would be good to know them? References will suffice, I can study them further.
Also, I guess, such optimization can use for solving the “optimal, paradox free deontic logic” as sketched in my previous question https://nforum.ncatlab.org/discussion/9838/category-of-institutions
I’m not sure there’s much more to say than what’s at universal construction, Kan extension, adjoint functor and related pages.
I wonder if with Simon Willerton’s treatment of the Legendre-Fenchel transform in enriched category theory, the description as ’optimization’ becomes literal. There’s a flavour of his idea here.
As for your logic case, sounds like you’re after some kind of initial object in a category of non-degenerate somethings, but conditions would have to be specified.
Adjunctions certainly crop up in logic. E.g., you can form a left and a right adjoint to the forgetful functor from the category of Boolean algebras to that of Heyting algebras (here).
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