Monoidal internalization

Comonoid object

Let be a monoidal category. A comonoid in is a monoid in the opposite category , #m/def/cat consisting of of the data

where is called the coünit and is called the comultiplication, and these satisfy the left/right coünit laws

and the coässociative law.

The category of comonoid objects is , which is simply the opposite category of .

Cocommutative cocomonoid

If is symmetric, a comonoid satisfying the cocommutative law

is called cocommutative.

Higher comultiplications

Note that by coässociativity, we can unambiguously define

Examples

• A comonoid in is a Bimonoid object (which is equivalently a monoid in ).
• A comonoid in is an -comonoid.

See also

• Comonoid morphism
• Tensor product of comonoids

#state/develop | #lang/en | #SemBr