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 set-theory monoid and a category-theory monoid are the same thing

References

• Category Theory 3.1 / Bartosz Milewski

Backlinks

• Monoid
  • A category with a single object is a monoid
• 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.