Portal:Category theory
Introduction
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups.
Several terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself.
Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations in 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.
Selected Article
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.
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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.
Categories
Selected Picture
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.
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 commutative 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?
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