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Dagger Category
In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution) is a category equipped with a certain structure called ''dagger'' or ''involution''. The name dagger category was coined by Peter Selinger. Formal definition A dagger category is a category \mathcal equipped with an involutive contravariant endofunctor \dagger which is the identity on objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an .... In detail, this means that: * for all morphisms f: A \to B, there exist its Hermitian adjoint, adjoint f^\dagger: B \to A * for all morphisms f, (f^\dagger)^\dagger = f * for all objects A, \mathrm_A^\dagger = \mathrm_A * for all f: A \to B and g: B \to C, (g \circ f)^\dagger = f^\dagger \circ g^\dagger: C \to A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. 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. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
