Monoid (category theory)

In category theory, a monoid (or monoid object) $(M,\mu,\eta)$ in a monoidal category $(\mathbf{C}, \otimes, I)$ is an object M together with two morphisms

$\mu�: M\otimes M\to M$ called multiplication,
and $\eta�: I\to M$ called unit,

such that the diagrams

and

commute. In the above notations, I is the unit element and $\alpha$ , $\lambda$ and $\rho$ are respectively the associativity, the left identity and the right identity of the monoidal category C.

Dually, a comonoid in a monoidal category C is a monoid in the dual category $\mathbf{C}^{\mathrm{op}}$ .

Suppose that the monoidal category C has a symmetry $\gamma$ . A monoid $M$ in C is symmetric when

$\mu\circ\gamma=\mu$ .

1 Examples
2 Categories of monoids
3 See also
4 References

Examples

A monoid object in Set (with the monoidal structure induced by the cartesian product) is a monoid in the usual sense.
A monoid object in Top (with the monoidal structure induced by the product topology) is a topological monoid.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton theorem.
A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the cartesian product) is a unital quantale.
A monoid object in (Ab, ⊗_Z, Z) is a ring.
For a commutative ring R, a monoid object in (R-Mod, ⊗_R, R) is an R-algebra.
A monoid object in K-Vect (again, with the tensor product) is a K-algebra, a comonoid object is a K-coalgebra.
For any category C, the category [C,C] of its endofunctors has a monoidal structure induced by the composition. A monoid object in [C,C] is a monad on C.

Categories of monoids

Given two monoids $(M,\mu,\eta)$ and $(M',\mu',\eta')$ in a monoidal category C, a morphism $f:M\to M'$ is a morphism of monoids when

$f\circ\mu = \mu'\circ(f\otimes f)$ ,
$f\circ\eta = \eta'$ .

The category of monoids in C and their monoid morphisms is written $\mathbf{Mon}_\mathbf{C}$ .

References

Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, Monoids, Acts and Categories (2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7

Monoid (category theory)

Contents

Examples

Categories of monoids

See also

References