Monoid (category theory)

In category theory, a monoid (or monoid object) (M,μ,η) 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 α, λ and ρ 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 γ. A monoid M in C is symmetric when

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

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,μ,η) and (M',μ',η') 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 $\mathbf{Mon}_\mathbf{C}$.

See also

• monoid (non-categorical definition)
• Act-S, the category of monoids acting on sets

References

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