Semigroup Object

A semigroup object in a Monoidal Category 𝒱 𝒱 \mathcal{V} is an interpretation of the theory of Semigroups into the Internal Language of 𝒱 𝒱 \mathcal{V} .

Explicitly, a semigroup object in 𝒱 𝒱 \mathcal{V} is an object S : 𝒱 : 𝑆 𝒱 S:\mathcal{V} equipped with a morphism m : S ⊗ S → S : 𝑚 → tensor-product 𝑆 𝑆 𝑆 m:S\otimes S\to S , such that the following associativity diagram commutes.

S⊗(S⊗S)tensor-product𝑆tensor-product𝑆𝑆{{S\otimes(S\otimes S)}}(S⊗S)⊗Stensor-producttensor-product𝑆𝑆𝑆{{(S\otimes S)\otimes S}}S⊗Stensor-product𝑆𝑆{{S\otimes S}}S⊗Stensor-product𝑆𝑆{{S\otimes S}}S𝑆{S}α𝛼\scriptstyle{\alpha}id⊗mtensor-productid𝑚\scriptstyle{\mathrm{id}\otimes m}m⊗idtensor-product𝑚id\scriptstyle{m\otimes\mathrm{id}}m𝑚\scriptstyle{m}m𝑚\scriptstyle{m}