## CATEGORY

### Category

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. 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 ...

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### category

#### Noun

1. A group, often named or numbered, to which items are assigned based on similarity or defined criteria.
This steep and dangerous climb belongs to the most difficult category.
I wouldn't put this book in the same category as the author's first novel.
2. A collection of objects, together with a transitively closed collection of composable arrows between them, such that every object has an identity arrow, and such that arrow composition is associative.
One well-known category has sets as objects and functions as arrows.
Just as a monoid consists of an underlying set with a binary operation "on top of it" which is closed, associative and with an identity, a category consists of an underlying digraph with an arrow composition operation "on top of it" which is transitively closed, associative, and with an identity at each object. In fact, a category's composition operation, when restricted to a single one of its objects, turns that object's set of arrows (which would all be loops) into a monoid.

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