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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 ...
Hybrid topology is also known as hybrid network. [19] Hybrid networks combine two or more topologies in such a way that the resulting network does not exhibit one of the standard topologies (e.g., bus, star, ring, etc.). For example, a tree network (or star-bus network) is a hybrid topology in which star networks are interconnected via bus ...
Many areas of computer science also rely on category theory, such as functional programming and semantics. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism. Metaphorically, a morphism is an arrow that maps its source to its target.
The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds , with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak ...
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 ...
The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n - Cat works fine because if a category C has finite products, the category of C -enriched categories has finite products too.
The monoidal product of two diagrams is represented by placing one diagram above the other. Indeed, all ZX-diagrams are built freely from a set of generators via composition and monoidal product, modulo the equalities induced by the compact structure and the rules of the ZX-calculus given below.