Portal:Category theory

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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.

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).

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Selected Biography

Samuel Eilenberg (born in Warsaw, September 30, 1913 and died in New York City, January 30, 1998) was a Polish and American mathematician. He spent much of his career in USA as a professor at Columbia University. His main interest was algebraic topology and foundational grounds to homology theory. He cofounded category theory with Saunders Mac Lane and wrote in 1965, Homological Algebra with Henri Cartan. Later, he worked in automata theory and pure category theory.


Selected Picture

Natural transformation.svg

A natural transformation, as a morphism between functors, respects the "functor structure". This idea, as usual in category theory, is expressed in a commutative diagram, pictured on the right.

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