Monoid (category theory)

 ( Monoidal Categories ) 

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object $M$ together with two

• $\backslash mu\backslash colon\; M\backslash otimes\; M\backslash to\; M$ called multiplication,
• $\backslash eta\backslash colon\; I\backslash to\; M$ called unit,

such that the pentagon diagram

and the unitor diagram

commute. In the above notation, $1$ is the identity morphism of $M$, $I$ is the unit element and $\backslash alpha,\backslash lambda$ and $\backslash rho$ are respectively the associativity, the left identity and the right identity of the monoidal category $\backslash mathcal\; C$.

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

Suppose that the monoidal category $\backslash mathcal\; C$ has a braiding $\backslash gamma$. A monoid $M$ in $\backslash mathcal\; C$ is commutative when .

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Examples

• A monoid object in Set, the category of sets (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense.
• A monoid object in Top, the category of topological spaces (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 argument.
• 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 , the category of abelian groups, is a ring.
• For a commutative ring R, a monoid object in
• , the category of modules over R, is a 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 (again, with the tensor product), is a unital associative K-algebra, and a comonoid object is a K-coalgebra.
• For any category C, the category of its 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 a terminal object and finite products, every object becomes a comonoid object via the diagonal morphism . Dually in a category with an initial object and Coproduct every object becomes a monoid object via .

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Categories of monoids Given two monoids and in a monoidal category C, a morphism is a morphism of monoids when

• f ∘ μ = μ′ ∘ ( f ⊗ f),
• f ∘ η = η′.

In other words, the following diagrams

,

commute.

The category of monoids in C and their monoid morphisms is written Mon _C.Section VII.3 in

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See also

• Act-S, the category of monoids acting on sets

• (2025). 9783110152487, Walter de Gruyter. ISBN 9783110152487

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