In category theory, an idempotent monad is, roughly speaking, a monad that theorizes a reflective subcategory.
Definition Let be a monad on a category . Then the following conditions are equivalent. The monad binary operation is a natural isomorphism. The forgetful functor for the Eilenberg–Moore category is full and faithful. For every -algebra the morphism is an isomorphism. Such a monad is called an idempotent monad.
Properties The notion of the idempotent monads is equivalent to that of the reflective subcategories. Given an idempotent monad on a category , the forgetful functor
from the Eilenberg–Moore category to is full and faithful. Conversely, every full and faithful right adjoint is monadic and its corresponding monad
is idempotent.
Idempotent monads associated with monads Let denote the category of monads on a category and their morphisms. Let denote the full subcategory of idempotnet monads. For a complete well-powered category , the inclusion functor
admits a right adjoint.
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