Internalization

Comonoid object

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

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

the coässociative law,

and optionally the cocommutative law (whence it is called cocommutative).

The category of comonoid objects is Category of comonoid objects, which is simply [[Category of monoid objects|]].

Higher comultiplications

Note that by coässociativity, we can unambiguously define

Examples

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

See also

• Tensor product of comonoids

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