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A monoidal category where every object has a left and right adjoint is called a rigid category. String diagrams for rigid categories can be defined as non-progressive plane graphs, i.e. the edges can bend backward. In the context of categorical quantum mechanics, this is known as the snake equation.
This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects. commutes. A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal ...
Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology. Categorical logic is now a well-defined field based on type theory for intuitionistic logics , with applications in functional programming and domain theory , where a cartesian closed category is taken as a non-syntactic ...
The category of small categories is a closed monoidal category in exactly two ways: with the usual categorical product and with the funny tensor product. [6] Given two categories and , let be the category with functors,: as objects and unnatural transformations: as arrows, i.e. families of morphisms {: ()} which do not necessarily satisfy the condition for a natural transformation.
In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing.
Quantum categories and quantum groupoids: A quantum category over a braided monoidal category V is an object R with an opmorphism h: R op ⊗ R → A into a pseudomonoid A such that h * is strong monoidal (preserves tensor product and unit up to coherent natural isomorphisms) and all R, h and A lie in the autonomous monoidal bicategory Comod(V ...
A braided monoidal category is a monoidal category equipped with a braiding—that is, a commutativity constraint that satisfies axioms including the hexagon identities defined below. The term braided references the fact that the braid group plays an important role in the theory of braided monoidal categories.
A braided monoidal functor is a monoidal functor between braided monoidal categories (with braidings denoted ) such that the following diagram commutes for every pair of objects A, B in : A symmetric monoidal functor is a braided monoidal functor whose domain and codomain are symmetric monoidal categories .