A monoid has an associative binary operation

Any monoid has an associative binary operation that takes any two elements in the associated set and produces an element in that set.

$$\otimes : a \rightarrow a \rightarrow a$$

The composition of morphisms in a monoidal category corresponds to an associative binary operation.

Backlinks

• The composition of morphisms in a monoidal category corresponds to an associative binary operation
  • For a monoidal category $M$ each morphism in $M(m, m)$ corresponds to the operation defined for the associated monoidal set on one of the elements.
• The identity morphism in a monoidal category corresponds to the unit element
  • Because the composition of morphisms in a monoidal category corresponds to an associative binary operation and composition with the identity morphism conforms to the law of identity, the unit element being used as an operand in the associative binary operation is the same thing as composition with the identity morphism.
• A set-theory monoid and a category-theory monoid are the same thing
  • The set theory definition and the category theory definition of monoids are different views over the same concept. Category theory defines a monoid as any category with a single object. Set theory defines a monoid as a set with an associative binary operation and a unit element.
• A monoid has a unit element
  • A monoid has a unit element $u$ which interacts with the associative binary operation defined for its set, $\\otimes$, in the following way