Search results
Results from the WOW.Com Content Network
The concept of monoidal category does not presume that values of such aggregate types can be taken apart; on the contrary, it provides a framework that unifies classical and quantum information theory. [1] In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the ...
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms. μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon diagram. and the unitor diagram commute.
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology . [ 1 ]
Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an -monoidal category. More generally, the center of a E k {\displaystyle E_{k}} -monoidal category is an algebra object in E k {\displaystyle E_{k}} -monoidal categories and therefore, by Dunn additivity , an E k ...
A closed monoidal category is a monoidal category such that for every object the functor given by right tensoring with . has a right adjoint, written ().This means that there exists a bijection, called 'currying', between the Hom-sets
Download as PDF; Printable version; ... Pages in category "Monoidal categories" ... Center (category theory) Closed monoidal category;
If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field. Given a field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the ...
In category theory, a branch of mathematics, a monad is a triple (,,) consisting of a functor T from a category to itself and two natural transformations, that satisfy the conditions like associativity.