Category theory MOC

Bimonoid object

A bimonoid 𝐡 in a Symmetric monoidal category is at once a monoid and comonoid in a compatible way, consisting of the data

πœ‚:1β†’π΅βˆ‡:π΅βŠ—π΅β†’π΅πœ–:𝐡→1Ξ”:π΅β†’π΅βŠ—π΅

such that it is a comonoid in [[Category of monoid objects|π–¬π—ˆπ—‡π–’]] or, what is the same, a monoid in π–¬π—ˆπ—‡π–’π¨π©. Unravelling definitions in terms of string diagrams, this gives the following compatibility conditions:

c

c

c

c

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