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Category theory

Commutative diagram for morphism.svg

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 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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Selected Biography

Saunders Mac Lane (4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. Their original motivation was homology theory and led to the formalization of what is now called homological algebra. His most recognized work in category theory is the textbook Categories for the Working Mathematician (1971).

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Functor cone (extended).svg

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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