Portal:Category theory

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Schematic representation of a category with objects X, Y, Z and morphisms f, g, gf. (The category's three identity morphisms 1X, 1Y and 1Z, if explicitly represented, would appear as three arrows, from the letters X, Y, and Z to themselves, respectively.)

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.

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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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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).


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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