Monoidal internalization

Bimonoid object

A bimonoid in a symmetric monoidal category is at once a monoid and comonoid in in a compatible way, #m/def/cat consisting of the data

such that it is a comonoid in or, what is the same, a monoid in .

String diagrams

Unravelling definitions in terms of string diagrams, this gives the following compatibility conditions:

c

c

c

c

The category of bimonoid objects is .

See also

#state/tidy | #lang/en | #SemBr