Monoid (category theory)

In , a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a (C, ⊗, I) is an M together with two morphisms μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon and the unitor diagram commute. In the above notation, 1 is the identity morphism of M, 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 Cop. Suppose that the monoidal category C has a γ. A monoid M in C is commutative when μ o γ = μ. A monoid object in , the (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense. A monoid object in , the (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 (, ⊗Z, Z), the , is a ring. For a commutative ring R, a monoid object in (, ⊗R, R), the over R, is a R-algebra. the category of graded modules is a graded R-algebra. the of R-modules is a differential graded algebra. A monoid object in K-Vect, the (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 [C,C] of its endofunctors has a monoidal structure induced by the composition and the identity functor IC. A monoid object in [C,C] is a on C. For any category with a terminal object and , every object becomes a comonoid object via the diagonal morphism . Dually in a category with an initial object and finite coproducts every object becomes a monoid object via . Given two monoids (M, μ, η) and (M', μ', η') in a monoidal category C, a morphism f : M → M ' is a morphism of monoids when f o μ = μ''' o (f ⊗ f), f o η = η.

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In mathematics, the category of rings, denoted by Ring, is the whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is , meaning that the class of all rings is proper. The category Ring is a meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure.

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In mathematics, in particular in homological algebra, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. TOC A differential graded algebra (or DG-algebra for short) A is a graded algebra equipped with a map which has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions: A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the .

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