Monoidal

Monoidal, in the context of category theory, refers to a property of a category equipped with a tensor product that combines two objects into a third, together with a distinguished unit object. A monoidal category consists of a category C, a bifunctor ⊗: C × C → C, a unit object I, and natural isomorphisms called the associator a_X,Y,Z: (X⊗Y)⊗Z → X⊗(Y⊗Z), and the left and right unitors l_X: I⊗X → X and r_X: X⊗I → X. These data satisfy the pentagon and triangle coherence axioms, which ensure that all ways of rebracketing or inserting the unit yield canonically isomorphic results.

Variants include strict monoidal categories, where the associator and unitors are identity morphisms, and symmetric or braided monoidal categories, which include a braiding or symmetry c_{X,Y}: X⊗Y → Y⊗X satisfying additional coherence conditions (hexagon axioms, and symmetry if applicable). Monoidal functors between monoidal categories preserve the tensor product up to specified coherent natural transformations; strong (or lax) monoidal functors give different levels of preservation of the tensor structure.

Examples include the category of vector spaces over a field with the usual tensor product and the field as unit object, and the category of abelian groups with the tensor product over the integers. Sets with the cartesian product form a symmetric monoidal category with the singleton as unit. Monoidal structures underpin concepts such as monoid objects, duals, and tensor categories, and they play a central role across algebra, topology, quantum algebra, and theoretical computer science.