A monad is a functor from a category to itself together with two natural transformations, commonly called μ (the "multiplication") and η (the "unit"), satisfying conditions that make μ monoidal and η an identity for it.
A monad $\Bbb T = (T,\eta,\mu)$ on a category $\mathcal C$ is a functor $T:\mathcal C \to \mathcal C$ together with natural transformations:
- $\eta : 1_{\mathcal C} \to T$, called the unit,
- $\mu : TT \to T$, called the multiplication,
such that:
- (left identity) $T\eta \circ \mu = 1_T$,
- (right identity) $\mu \circ \eta_T = 1_T$,
- (associativity) $\mu \circ T\mu = \mu \circ \mu_T$.