nLab comonoid



Monoid theory


Category theory



A comonoid (or comonoid object) in a monoidal category \mathcal{M} is a monoid object CC in the opposite category op\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 CC 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 CC is in addition equipped with a morphism η:IC\eta \colon I \rightarrow C

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


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.

Last revised on November 25, 2022 at 05:51:24. See the history of this page for a list of all contributions to it.