Monoidal internalization

Bimonoid object

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

𝜂:1→𝐵∇:𝐵⊗𝐵→𝐵𝜖:𝐵→1Δ:𝐵→𝐵⊗𝐵

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

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 𝖡𝗂𝗆𝗈𝗇𝖢.

See also

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