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Related concepts (35)
Exponential object
In mathematics, specifically in , an exponential object or map object is the generalization of a function space in set theory. with all and exponential objects are called . Categories (such as of ) without adjoined products may still have an exponential law. Let be a category, let and be of , and let have all with .
Cartesian closed category
In , a is Cartesian closed if, roughly speaking, any morphism defined on a of two can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by , whose internal language, linear type systems, are suitable for both quantum and classical computation.
Direct image functor
In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F.
Subobject classifier
In , a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.
Functor category
In , a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, is another object in the category). Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the whose s are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of 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 .
Functor represented by a scheme
In algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is (up to natural bijections) the set of all morphisms . The scheme X is then said to represent the functor and that classify geometric objects over S given by F. The best known example is the Hilbert scheme of a scheme X (over some fixed base scheme), which, when it exists, represents a functor sending a scheme S to a flat family of closed subschemes of .
Presheaf (category theory)
In , a branch of mathematics, a presheaf on a is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on into a category, and is an example of a . It is often written as . A functor into is sometimes called a profunctor.
Yoneda lemma
In mathematics, the Yoneda lemma is arguably the most important result in . It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the of any into a (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category.
Equivalence of categories
In , a branch of abstract mathematics, an equivalence of categories is a relation between two that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned.