Idempotent monad

In category theory, an idempotent monad is, roughly speaking, a monad that theorizes a reflective subcategory.

Definition

Let $(T, η, μ)$ be a monad on a category $𝒞$ . Then the following conditions are equivalent.

The monad binary operation $μ : T T \Rightarrow T$ is a natural isomorphism.
The forgetful functor $\begin{array}{l} 𝒞_{T} \to 𝒞 \\ (A, e) \mapsto A \\ (f : (A, e) \to (A^{'}, e^{'})) \mapsto (f : A \to A^{'}) \end{array}$ for the Eilenberg–Moore category is full and faithful.
For every $T$ -algebra $(A, e)$ the morphism $e : T A \to A$ 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 $𝒞$ , the forgetful functor

𝒞_{T} \to 𝒞

from the Eilenberg–Moore category $𝒞_{T}$ to $𝒞$ is full and faithful. Conversely, every full and faithful right adjoint $I : 𝒟 \to 𝒞$ is monadic and its corresponding monad

(I \circ R, η, I \circ ϵ \circ R)

is idempotent.^[1]^{: 197, Corollary 4.2.4}