Monoid object

Summary

A monoid object in a monoidal category is an object $M$ with a multiplication $M \otimes M \to M$ and a unit $I \to M$. These must satisfy the associativity and unit laws, written as equations between morphisms. It generalizes the notion of a monoid to any monoidal category. Monoids in sets are ordinary monoids, monoids in abelian groups are rings, monoids in vector spaces are associative algebras, and monoids among endofunctors are monads. Many algebraic structures are thus one notion read in different monoidal categories.