co-unitality in nLab

Contents

Contents

Idea

The formal dual of unitality:

An object in a monoidal category which is equipped with a comultiplication map which satisfies both co-unitality and co-associativity is called a co-monoid.

Definition

Given a monoidal category (𝒞,⊗)(\mathcal{C}, \otimes) and an object AA in 𝒞\mathcal{C} equipped with a morphism (“co-multiplication”) Δ:A⟶A⊗A\Delta \colon A \longrightarrow A \otimes A, and a morphism ϵ:A⟶I\epsilon \colon A \longrightarrow I, ϵ\epsilon is called a counit if the following diagrams commute

where II is the unit of the monoidal category and λ,ρ\lambda, \rho are the left and right unitors respectively.

• associativity, co-associativity