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Daniel Teixeira (Oct 04 2024 at 01:51):Is a natural transformation between equivalences necessarily invertible?
Mike Shulman (Oct 04 2024 at 02:19):No.
Mike Shulman (Oct 04 2024 at 02:20):For example, when a monoid is regarded as a 1-object category, a natural transformation from its identity functor to itself is the same as an element of the center of the monoid.
John Baez (Oct 04 2024 at 06:31):For another example, the identity functor from (the category of vector spaces and linear maps) to itself is an equivalence. But there's a natural transformation from this identity functor to itself that sends any vector space to the zero map , and this is not invertible since multiplying by zero is not invertible.
Daniel Teixeira (Oct 06 2024 at 14:13):fair enough, endomorphisms save the day again
Daniel Teixeira (Oct 06 2024 at 14:16):A friend challenged me to cook up a 2-category with invertible 1- but not 2-cells, hence the question. I couldn't find any such natural transformation, so the example instead was very much similar to John's: take the full subcategory of , at the 1-d vector spaces, deloop it. Now every morphism is invertible, and so is almost every 2-morphism - except for the zero maps.