Algebraically idempotent relative monads, where every algebra is idempotent, recover the classical characterization that the forgetful functor to the ambient category is reflective.
‘Monads need not be endofunctors’
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Idempotence for relative monads
Algebraically idempotent relative monads, where every algebra is idempotent, recover the classical characterization that the forgetful functor to the ambient category is reflective.