Monoidale

Monoidale is a term used in category theory to refer to a monoidal category, i.e., a category equipped with a tensor product, a unit object, and coherence isomorphisms that generalize the notion of multiplication and unit from algebra to categories. In a monoidal category (C, ⊗, I), there are natural isomorphisms called the associator α, and the left and right unitors λ and ρ, which satisfy the pentagon and triangle coherence conditions. If these constraints are identities, the category is called strict; if the tensor product is commutative up to a natural isomorphism, the structure is symmetric monoidal, and if it is braiding-compatible, it is braided monoidal.

Within a monoidal category, one can also define monoid objects: an object M together with a multiplication μ: M ⊗ M → M and a unit η: I → M that satisfy associativity and unit laws expressed by commutative diagrams. Dually, comonoids are defined with comultiplication and counit. Such monoid objects generalize the notion of monoids in algebra to the categorical setting, enabling notions of algebras and modules internal to a monoidal category.

In higher category theory, a monoidale (often called a pseudomonoid) is a monoid object in a monoidal bicategory. For example, in the monoidal bicategory Cat, monoidales correspond to ordinary monoidal categories. Monoidales are relevant in enriched category theory, quantum algebra, and representation theory, where the interplay between objects and their tensor products encodes algebraic structure at a higher level.

Common examples include the category of sets with Cartesian product (Set, ×, 1) and the category of vector spaces over a field with the tensor product (Vect_k, ⊗, k). Monoidales provide a unifying framework for "multiplication" across objects and morphisms.