Examples
- The monoidal category Set of sets and functions, with cartesian product as the tensor product, is a closed monoidal category. Here, the internal hom is the set of functions from to . In computer science, the bijection between tensoring and the internal hom is known as currying, particularly in functional programming languages. Indeed, some languages, such as Haskell and Caml, explicitly use an arrow notation to denote a function. This example is a Cartesian closed category.
- More generally, every Cartesian closed category is a symmetric monoidal closed category, when the monoidal structure is the cartesian product structure. Here the internal hom is usually written as the exponential object .
- The monoidal category FdVect of finite-dimensional vector spaces and linear maps, with its usual tensor product, is a closed monoidal category. Here is the vector space of linear maps from to . This example is a compact closed category.
- More generally, every compact closed category is a symmetric monoidal closed category, in which the internal Hom functor is given by .
Read more about this topic: Closed Monoidal Category
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