Monoid

A monoid is an algebraic structure with an identity value and an associative binary operation.

• A category with a single object is a monoid

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• Monoidal Identity
  • The monoidal identity is the identity value for a particular monoid. A monoid must have an identity value and an associative binary operation. The identity value $m$ is a value which makes no change to any other value $n$ in that monoid, when $n$ is combined with $m$ using that binary operation.
• A category with a single object is a monoid
  • Any category with a single object is a monoid. The object in a monoidal category can have any number of morphisms greater than 1 - the requirement for an identity morphism isn't relaxed.
• 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.
• 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
  • The unit element defined for a set-theory monoid corresponds to the identity morphism within a monoidal category.
• 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