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Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942-1945, in connection with algebraic topology.

The term "abstract nonsense" has been used by some critics to refer to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting calculations in abstract algebra. Diagram chasing is a visual method of arguing with abstract 'arrows'. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

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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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Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered to be one of the greatest mathematicians of the 20th century. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and was co-awarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize on ethical grounds in an open letter to the media. His work in algebraic geometry led to considerable developments in category theory, such as the concept of Abelian category and derived category.

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In category theory, a limit of a diagram is defined as a cone satisfying a universal property. Products and equalizers are special cases of limits. The dual notion is that of colimit.

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Did you know?

... that in higher category theory, there are two major notions of higher categories, the strict one and the weak one ?
... that factorization systems generalize the fact that every function is the composite of a surjection followed by an injection ?
... that in a multicategory, morphisms are allowed to have a multiple arity, and that multicategories with one object are operads ?
... that it is possible to define the end and the coend of certain functors ?
... that in the category of rings, the coproduct of two rings is their tensor product ?
... that the Yoneda lemma proves that any small category can be embedded in a presheaf category ?
... that it is possible to compose profunctors so that they form a bicategory?