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Nathanael Arkor (May 12 2023 at 23:53):I remember seeing a slick proof somewhere that the unit of an adjunction in a 2-category is invertible if and only if there is some isomorphism (which then implies there is a canonical isomorphism). Does anyone have an idea where such a proof might be found?
John Baez (May 13 2023 at 01:24):This nLab section gives a bunch of equivalent characterizations of adjunctions where the counit is an isomorphism, and by taking "op" these should give characterizations of adjunctions where the unit is an isomorphism. However, none of them involve mere existence of some isomorphism, so doesn't really answer your question.
Mike Shulman (May 13 2023 at 01:29):It's Lemma A1.1.1 in Sketches of an Elephant (written in Cat, but the proof works in any 2-category). The argument is to transfer the monad structure on across the isomorphism to a monad structure on , and observe that an identity morphism has a unique monad structure, which is idempotent; hence the original monad structure on was also idempotent.
John Baez (May 13 2023 at 06:34):I'm having trouble observing that an identity morphism in any 2-category has a unique monad structure. When I look at the monad axioms and set the monad to an identity, I seem to just get the unit of the monad is the left and right inverse of its multiplication. What am I doing wrong?
Morgan Rogers (he/him) (May 13 2023 at 08:11):You don't need uniqueness (I don't think it holds). What you've said ensures that the transferred monad (and hence the original monad) is idempotent.
Nathanael Arkor (May 13 2023 at 08:33):Mike Shulman said:
It's Lemma A1.1.1 in Sketches of an Elephant (written in Cat, but the proof works in any 2-category). The argument is to transfer the monad structure on across the isomorphism to a monad structure on , and observe that an identity morphism has a unique monad structure, which is idempotent; hence the original monad structure on was also idempotent.
Perfect, thank you very much!
Nathanael Arkor (May 13 2023 at 08:38):For anyone else who is interested in the proof:
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Notification Bot (May 13 2023 at 12:41):Nathanael Arkor has marked this topic as resolved.
John Baez (May 13 2023 at 15:29):I added this to the list of equivalent characterization of reflective subcategories on the nLab. For some reason I'm getting into learning category theory by adding results to the nLab. Even when forget what I learn the world will remember. :upside_down:
Mike Shulman (May 13 2023 at 16:16):I don't know why I thought the monad structure on an identity morphism is unique. I guess what's true is that a monad structure on an identity morphism is the same as an automorphism of it. And that any monad structure on an identity morphism is isomorphic, as a monad, to the trivial monad -- but the isomorphism isn't in general an identity 2-cell.
John Baez (May 13 2023 at 17:45):I decided you were using "unique" in some modern univalent sense. :upside_down: But then I got curious about the automorphism group of an endofunctor and how it acts on the set of monad structures on this endofunctor. Like: does this say anything interesting about monads, or is it just boring?
Mike Shulman (May 13 2023 at 22:42):It's not even true in the univalent sense that there's a unique monad structure on an identity morphism, but I guess we can say that there's a unique monad whose underlying morphism is (merely) an identity morphism. (-:
Mike Shulman (May 13 2023 at 22:44):As for your question, well, the automorphism group of an X always acts on the set of Y-structures on that X. The automorphism group of a set acts on the set of group structures on that set, etc. Nothing particularly interesting about the monad case jumps to mind.