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Oscar Cunningham (Apr 21 2023 at 17:56):What's an adjunction in the 2-category of sets, spans and morphisms or spans? Does it correspond to some existing notion?
Mike Shulman (Apr 21 2023 at 18:00):It's a function!!
John Baez (Apr 21 2023 at 19:41):This is an instance of a general pattern! An adjunction in the 2-category of sets, relations and implications between relations is also a function. An adjunction in the 2-category of categories, profunctors and natural transformations between profunctors is a functor (from one category to the Cauchy completion of the other).
It's the left adjoint that "is" the function or functor here, so sometimes people call left adjoints in bicategories "maps".
John Baez (Apr 21 2023 at 19:42):I'm sure some other people here know a lot more about this pattern than I do.
Oscar Cunningham (Apr 21 2023 at 20:00):Thank you Mike!
Oscar Cunningham (Apr 21 2023 at 20:01):Yes, I'd seen that thing about Cauchy completions before. I always thought it was a bit weird. In the same way that a displayed category is a lax functor into Span, but a lax normal functor into Prof.
Oscar Cunningham (Apr 21 2023 at 20:02):So I was hoping that similarly the adjunctions in Span turned out to be less weird than the adjunctions in Prof.
Mike Shulman (Apr 21 2023 at 20:22):Basically the point is that discrete categories are always Cauchy complete.
John Baez (Apr 21 2023 at 21:43):The Cauchy completion stuff here seemed weird to me too at first:
An adjunction in the 2-category of categories, profunctors and natural transformations between profunctors is a functor (from one category to the Cauchy completion of the other).
John Baez (Apr 21 2023 at 21:48):But it helped me to realize that presheaves on a category are the same as presheaves on its Cauchy completion, and a profunctor is just a presheaf with extra structure, so it would be extremely optimistic to think that a profunctor can be expected to be "smart enough" to know whether it's supposed to be going to some category or to its Cauchy completion. When you try to take a left adjoint profunctor from C to D and turn it into a functor, it might fail because it's trying to map an object of C to a 'summand' of an object of D, i.e. a nonexistent object that pops into existence when you split idempotents, as you do when taking the Cauchy completion of D.