Bimonoid object

Category of bimonoid objects

The category of bimonoids in $𝖢$ , denoted $𝖡 𝗂 𝗆 𝗈 𝗇 𝖢$ , is a category where an object is a bimonoid object in $𝖢$ and a morphism is a bimonoid morphism. #m/def/cat We therefore have isomorphisms of categories

𝖡 𝗂 𝗆 𝗈 𝗇 𝖢 ≅ 𝖢 𝗈 𝗆 𝗈 𝗇 𝖬 𝗈 𝗇 𝖢 ≅ 𝖬 𝗈 𝗇 𝖢 𝗈 𝗆 𝗈 𝗇 𝖢

where we invoke $𝖬 𝗈 𝗇 𝖢$ and $𝖢 𝗈 𝗆 𝗈 𝗇 𝖢$ .

As the bicartesian completion of a symmetric monoidal category

If $𝖢$ is a symmetric monoidal category then the tensor product of two bimonoids in $𝖢$ naturally carries the structure of a bimonoid.

By the above isomorphism, we have the following diamond of forgetful functors which are all monoidal if $𝖢$ is symmetric.

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