nLab
unit of a monad

Contents

Defintion

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

ϵ:id 𝒞T \epsilon \;\colon\; id_{\mathcal{C}} \to T

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

ϵ X:XT(X) \epsilon_X \;\colon\; X \to T(X)

in 𝒞\mathcal{C}.

Dually, there is a counit of a comonad.

Properties

Relation to adjunctions

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

Revised on December 5, 2013 05:52:42 by Urs Schreiber (89.204.138.93)