# Portal:Category theory

## Introduction

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.

## Selected Article

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.

## Selected Biography

Alexander Grothendieck (March 28, 1928 Berlin, Germany - November 13, 2014) is considered to be one of the greatest mathematicians of the 20th century. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and was co-awarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize on ethical grounds in an open letter to the media. His work in algebraic geometry led to considerable developments in category theory, such as the concept of Abelian category and derived category.

## Mathematics-related portals

 Logic Mathematics Numbertheory Physics Science Set theory Statistics

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