MonoidalKategorie

A monoidal category is a category equipped with a tensor product and a unit object. The tensor product is a bifunctor, denoted by $\otimes$, that takes two objects from the category and produces a new object. This tensor product is associative up to a natural isomorphism. Specifically, for any three objects $A$, $B$, and $C$ in the category, there exists an isomorphism $\alpha_{A,B,C} : (A \otimes B) \otimes C \to A \otimes (B \otimes C)$. This isomorphism must satisfy the pentagon identity, which ensures a consistent way of re-associating tensor products of multiple objects.

In addition to the tensor product, a monoidal category has a unit object, often denoted by $I$. This unit object acts as an identity for the tensor product. For any object $A$ in the category, there are natural isomorphisms $\lambda_A : I \otimes A \to A$ and $\rho_A : A \otimes I \to A$. These isomorphisms, along with the associators $\alpha$, must satisfy the triangle identities, which relate the unitors to the associators.

Monoidal categories are fundamental structures in category theory and have applications in various fields, including abstract algebra, functional programming, and theoretical physics. They generalize the notion of monoids. A monoid can be viewed as a monoidal category with a single object, where the tensor product corresponds to the monoid multiplication and the unit object corresponds to the monoid identity element. Examples of monoidal categories include the category of vector spaces with tensor product, the category of sets with the Cartesian product, and categories of modules over a ring. The structure provided by the tensor product and unit object allows for the development of algebraic theories within these categories.