Monoidal internalization

Semigroup object

Let (𝖢, ⊗,𝟙,𝛼,𝜆,𝜌) be a monoidal category. A semigroup in 𝖢 consists of the data #m/def/cat

𝑀⊗𝑀𝑚⟶𝑀

where 𝑚 is called the multiplication, and these satisfy the associative law. Moreover, if we are in a Symmetric monoidal category with braiding 𝜏, then (𝑀,𝑚,𝑒) is called commutative iff it satisfies the commutative law.

Commutative diagrams

Associative law: https://q.uiver.app/#q=WzAsNSxbMCwwLCIoTSBcXG90aW1lcyBNKSBcXG90aW1lcyBNIl0sWzIsMCwiTSBcXG90aW1lcyAoTSBcXG90aW1lcyBNKSJdLFsxLDIsIk0gXFxvdGltZXMgTSJdLFszLDIsIk0iXSxbNCwwLCJNIFxcb3RpbWVzIE0iXSxbMCwxLCJcXGFscGhhIl0sWzAsMiwibSBcXG90aW1lcyAxIiwyXSxbMiwzLCJtIiwyXSxbMSw0LCIxIFxcb3RpbWVzIG0iXSxbNCwzLCJtIl1d

Commutative law: https://q.uiver.app/#q=WzAsMyxbMCwwLCJNIFxcb3RpbWVzIE0iXSxbMiwwLCJNIFxcb3RpbWVzIE0iXSxbMSwxLCJNIl0sWzAsMiwibSIsMl0sWzEsMiwibSJdLFswLDEsIlxcdGF1Il1d

String diagrams

Associativity: c

Commutativity: c

We can thence define a Semigroup morphism and 𝖲𝗆𝗀𝗋𝖢. These concepts admit duals, see Cosemigroup object.

#state/develop | #lang/en | #SemBr