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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.
In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's binary operation is continuous. Every topological group is a topological monoid.
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 ...
such that the pentagon diagram. and the unitor diagram commute. In the above notation, 1 is the identity morphism of M, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category C op.
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms.
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.
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.
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.