Monoid (category theory)

• μ: M ⊗ M → M called multiplication,
• η: I → M called unit,

Examples

• A monoid object in Set, the category of sets, is a monoid in the usual sense.
• A monoid object in Top, the category of topological spaces, is a topological monoid.
• A monoid object in the category of monoids is just a commutative monoid. This follows easily from the Eckmann–Hilton argument.
• A monoid object in the category of complete join-semilattices Sup is a unital quantale.
• A monoid object in, the category of abelian groups, is a ring.
• For a commutative ring R, a monoid object in
• *, the category of modules over R, is an R-algebra.
• * the category of graded modules is a graded R-algebra.
• * the category of chain complexes of R-modules is a differential graded algebra.
• A monoid object in K-Vect, the category of K-vector spaces, is a K-algebra, and a comonoid object is a K-coalgebra.
• For any category C, the category of its endofunctors has a monoidal structure induced by the composition and the identity functor I_C. A monoid object in is a monad on C.
• For any category with finite products, every object becomes a comonoid object via the diagonal morphism. Dually in a category with finite coproducts every object becomes a monoid object via.

  Categories of monoids

• f o μ = μ o,
• f o η = η.