Monoid (category theory)

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In category theory, a monoid (or monoid object) $(M,μ,η)$ in a monoidal category C 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

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

[edit] 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.