Definition 5.3 (Eilenberg-Moore algebra). Let $𝕋 = (T, η, μ)$ be a monad on $C$ . By an Eilenberg-Moore algebra for $𝕋$ we mean a pair $(A, α)$ where $A \in ob C$ and $α : T A \to A$ satisfies

A TA TTA T A

(4) : ηA1Aα A(5) : TμαααA TA A

A homomorphism

f : (A, α) \to (B, β)

is a morphism

f : A \to B

satisfying

We write

C^{𝕋}

for the category of

𝕋

-algebras and homomorphisms.