nLab
comonoid
Contents
Context
Monoid theory
monoid theory in algebra :

monoid , infinity-monoid

monoid object , monoid object in an (infinity,1)-category

Mon , CMon

monoid homomorphism

submonoid , quotient monoid?

divisor , multiple? , quotient element?

inverse element , unit , irreducible element

ideal in a monoid

principal ideal in a monoid

commutative monoid

cancellative monoid

GCD monoid

unique factorization monoid

Bézout monoid

principal ideal monoid

group , abelian group

absorption monoid

free monoid , free commutative monoid

graphic monoid

monoid action

module over a monoid

localization of a monoid

group completion

endomorphism monoid

super commutative monoid

Algebra
Category theory
category theory

Concepts
Universal constructions
Theorems
Extensions
Applications
Contents
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.

Last revised on November 25, 2022 at 05:51:24.
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