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Paolo Perrone (Jan 20 2022 at 15:43):It is well known that there is a bijection between morphisms of monads and (backward) functors between the Eilenberg-Moore categories commuting with the forgetful functors.
Is a similar statement known for Kleisli categories, does anyone know?
Nathanael Arkor (Jan 20 2022 at 16:27):Yes. Monads morphisms are in bijection with functors between Kleisli categories commuting with the inclusion functors.
Nathanael Arkor (Jan 20 2022 at 16:27):It's formally dual to the statement about EM categories.
Nathanael Arkor (Jan 20 2022 at 16:29):The earliest reference is most likely Maranda's On fundamental constructions and adjoint functors, though it's also implicit in Linton's work.
Oscar Cunningham (Jan 20 2022 at 16:46):BY 'inclusion' functors here do you mean the free functors? I think that's what it would be if it was dual to the EM result.
Nathanael Arkor (Jan 20 2022 at 16:53):The functors into the Kleisli categories, yes.
Paolo Perrone (Jan 21 2022 at 09:03):Thank you!