# Category (mathematics)

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In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may be abstract entities of any kind. Categories generalize monoids, groupoids and preorders. In addition, 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 and the list of category theory topics.

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. Many well-known categories are conventionally identified by a short capitalized word or abbreviation in bold or italics such as Set (category of sets and set functions),[1] Ring (category of rings and ring homomorphisms),[2] or Top (category of topological spaces and continuous maps).[3]

## Contents

### Definition

A category C consists of