Portal:Category theory/Selected article

Selected article

In mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "arrows" of any kind that have a few basic properties (the ability to compose the arrows associatively and the existence of an identity arrow for each object). Many well-known categories are conventionally identified by a short capitalized word or abbreviation in bold or italics such as Set (category of sets) or Ring (category of rings).

Selected article

In category theory, a functor is a special type of mapping between categories. Functors respect the "category structure": they send an identity to an identity and preserves the composition. Functors are common in mathematics and arise in different kinds: faithful, exact, adjoint. Sheaves are special contravariant functors from the partially ordered set of open sets of a topological space to a complete category. Functors are the morphisms in the category of small categories.

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In category theory, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most basic notions of categorical algebra and consequently appear in the majority of its applications.

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In category theory, a topos (plural "topoi" or "toposes") is a type of category that behaves like the category of sheaves of sets on a topological space, or more generally, like the category of sheaves on some site. The origin of Topos theory is, for the most part, found in algebraic topology. The theory has since known a considerable development and has applications in various fields, it also became a fundamental component of categorical logic.

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In mathematics, an Abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative to unify several cohomology theories by Alexander Grothendieck and has major applications in algebraic geometry, cohomology and pure category theory.

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In category theory, the derived category of an Abelian category is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors. The development of the theory, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s. Derived categories have since appeared outside of algebraic geometry, for example in D-modules theory and microlocal analysis.