nLab comonoid

Definition

A comonoid (or comonoid object) in a monoidal category $\mathcal{M}$ is a monoid object $C$ in the opposite category $\mathcal{M}^{op}$ (which canonically becomes a monoidal category via the same tensor product operation as in $\mathcal{M}$).

With the usual definition of monoids as having a unit, this means that a comonoid $C$ is equipped with a counit, which in string diagram-notation for $\mathcal{M}$ is of this form:

One speaks of a unital comonoid (or unital comonoid object; internal to Vect also called an augmented coalgebra) if $C$ is in addition equipped with a morphism $\eta \colon I \rightarrow C$

which verifies the properties that the unit would verify if $C$ was a bimonoid, ie. in string diagrams:

Examples

For example, a comonoid in Vect (with its usual tensor product) is called a coalgebra. Every set can be made into a comonoid in Set (with the cartesian product) in a unique way. More generally, every object in a cartesian monoidal category can be made into a comonoid in a unique way.

Related concepts

• coalgebra

• co-associativity

• co-unitality