Portal:Category theory

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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 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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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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Categories

Homological algebra • Additive categories
Duality theories • Sheaf theory
Higher category theory • Monoidal categories
Categorical logic • Topos theory
Category-theoretic categories
Objects • Functors
Category theorists

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In homological algebra, the snake lemma, a statement valid in every abelian category, is the crucial tool used to construct the long exact sequences.

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Topics

Homological algebra: Abelian category • Sheaf theory • K-theory

Topos theoryEnriched category theoryHigher category theory

  Monoidal category • Closed category • Dagger category

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