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

$$\exists u \forall a. u \otimes a = a \otimes u = a$$

The identity morphism in a monoidal category corresponds to the unit element

Backlinks

• 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.