nLab co-unitality

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 $(\mathcal{C}, \otimes)$ and an object $A$ in $\mathcal{C}$ equipped with a morphism (“co-multiplication”) $\Delta \colon A \longrightarrow A \otimes A$, and a morphism $\epsilon \colon A \longrightarrow I$, $\epsilon$ is called a counit if the following diagrams commute

where $I$ is the unit of the monoidal category and $\lambda, \rho$ are the left and right unitors respectively.

Related concepts

• associativity, co-associativity