# 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, 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).

## Categories

## Selected Picture

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.

## 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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