monoid theory in algebra:
symmetric monoidal (∞,1)-category of spectra
A comonoid (or comonoid object) in a monoidal category is a monoid object in the opposite category (which canonically becomes a monoidal category via the same tensor product operation as in ).
With the usual definition of monoids as having a unit, this means that a comonoid is equipped with a counit, which in string diagram-notation for is of this form:
One speaks of a unital comonoid (or unital comonoid object; internal to Vect also called an augmented coalgebra) if is in addition equipped with a morphism
which verifies the properties that the unit would verify if 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.