Category of monoid objects

The category of monoids in $𝖢$ , denoted $𝖬 𝗈 𝗇 𝖢$ , is a category where an object is a monoid in $𝖢$ and a morphism is a Monoid morphism. #m/def/cat In particular, $𝖬 𝗈 𝗇 𝖲 𝖾 𝗍$ is the same as $𝖬 𝗈 𝗇$ .

As the cocartesian completion of a symmetric monoidal category

If $𝖢$ is a symmetric monoidal category then the tensor product of two monoids in $𝖢$ naturally carries the structure of a monoid with the unit and multiplication

With the product defined as such, $𝖬 𝗈 𝗇 𝖢$ is in fact a cocartesian category. #m/thm/cat

Proof

Suppose $𝑅, 𝐵, 𝑃$ are monoids in $𝖢$ . We define monoid morphisms

and claim these satisfy the universal property of the coproduct. Namely, if $𝑓 : 𝑅 \to 𝑃$ and $𝑔 : 𝐵 \to 𝑃$ are monoid morphisms in $𝖢$ , then we claim ${𝑓, 𝑔}$ is the unique monoid morphism such that

commutes. Suppose that $ℎ : 𝑅 \otimes 𝐵 \to 𝑃$ can replace ${𝑓, 𝑔}$ in the commutative diagram. Then

so $ℎ = {𝑓, 𝑔}$ as required.

There is an evident forgetful functor $𝖬 𝗈 𝗇 𝖢 \to 𝖢$ , monoidal if $𝖢$ is symmetric.
$𝖢 𝗈 𝗆 𝗈 𝗇 𝖢$ = $𝖬 𝗈 𝗇 𝖢 𝐨 𝐩 𝐨 𝐩$

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

Category of monoid objects

As the cocartesian completion of a symmetric monoidal category

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