symmetric monoidal (∞,1)-category of spectra
A commutative monoid is a monoid where the multiplication satisfies the commutative law:
Alternatively, just as a monoid can be seen as a category with one object, a commutative monoid can be seen as a monoidal category with one object and one morphism.
Commutative monoids with homomorphisms between them form a category of commutative monoids.
An abelian group is a commutative monoid that is also a group.
The natural numbers (together with 0) form a commutative monoid under addition.
Every bounded semilattice is an idempotent commutative monoid, and every idempotent commutative monoid yields a semilattice, (see that entry).