Karl Jones at 11:55, 16 September 2016

2016-09-16T11:55:58Z

Karl Jones: Created page with "In mathematics, a '''category''' is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to c..."

2016-09-14T04:04:40Z

Created page with "In mathematics, a '''category''' is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to c..."

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In [[mathematics]], a '''category''' is an [[algebraic structure]] that comprises "objects" that are linked by "arrows".

A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object.

== Description ==

A simple example is the category of sets, whose objects are sets and whose arrows are functions. On the other hand, any monoid can be understood as a special sort of category, and so can any preorder. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships. This is the central idea of category theory, a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent.

Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics.

For more extensive motivational background and historical notes, see [[category theory]].

Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two categories may also be considered "equivalent" for purposes of category theory, even if they are not precisely the same.

Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include:

* [[Category of sets|Set]], the category of [[Set (mathematics)|sets]] and set functions;
* [[Category of rings|Ring]], the category of [[Ring (mathematics)|rings]] and [[Ring homomorphism|ring homomorphisms]]; and
* [[Category of topological spaces|Top]], the category of [[Topological space|topological spaces]] and [[Continuous map|continuous maps]].

All of the preceding categories have the identity map as identity arrow and composition as the associative operation on arrows.

The classic and still much used text on [[category theory]] is ''Categories for the Working Mathematician'' by [[Saunders Mac Lane]].

== See also ==

* [[Category theory]]
* [[Enriched category]]
* [[Higher category theory]]
* [[Quantaloid]]
* [[Table of mathematical symbols]]

== External links ==

* [https://en.wikipedia.org/wiki/Category_(mathematics) Category (mathematics)] @ Wikipedia.org

[[Category:Mathematics]]

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 * [[Enriched category]]
 * [[Higher category theory]]
 * [[Quantaloid]]
 * [[Table of mathematical symbols]]

Category (mathematics) - Revision history

Karl Jones at 11:55, 16 September 2016

Karl Jones: Created page with "In mathematics, a '''category''' is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to c..."