nLab unit of a monad

Defintion

Given a monad $T \;\colon\; \mathcal{C} \to \mathcal{C}$, its unit is the natural transformation

$$\epsilon \;\colon\; id_{\mathcal{C}} \to T$$

which is part of the definition of monad. Hence for every object $X \in \mathcal{C}$ the component of the unit on $X$ is a morphism

$$\epsilon_X \;\colon\; X \to T(X)$$

in $\mathcal{C}$.

Dually, there is a counit of a comonad.

Properties

Relation to adjunctions

If $(L \dashv R)$ is an adjunction that gives rise to the monad $T$ as $T \simeq R \circ L$, then the unit of the monad is equivalently the unit of the adjunction.

Related concepts

• unit

  • unit object

  • unit of an adjunction

  • unit of a monad