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In category theory, a branch of mathematics, the opposite category or dual category C op of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself.
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category C op.Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite ...
The connection of generalization to specialization (or particularization) is reflected in the contrasting words hypernym and hyponym.A hypernym as a generic stands for a class or group of equally ranked items, such as the term tree which stands for equally ranked items such as peach and oak, and the term ship which stands for equally ranked items such as cruiser and steamer.
Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction. The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. [2]
In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
For example, if is a field, then for every vector space over we have a "natural" injective linear map from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor.
For example, if some property P(x,y) of real numbers is known to be symmetric in x and y, namely that P(x,y) is equivalent to P(y,x), then in proving that P(x,y) holds for every x and y, one may assume "without loss of generality" that x ≤ y.
Examples include quotient spaces, direct products, completion, and duality. 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 ...