Category (noun)
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."
Category (noun)
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."

Wiktionary

Category (noun)
a class or division of people or things regarded as having particular shared characteristics
"the various categories of research"
Category (noun)
each of a possibly exhaustive set of classes among which all things might be distributed.
Category (noun)
each of the a priori conceptions applied by the mind to sense impressions.

Oxford Dictionary

Category (noun)
One of the highest classes to which the objects of knowledge or thought can be reduced, and by which they can be arranged in a system; an ultimate or undecomposable conception; a predicament.
Category (noun)
Class; also, state, condition, or predicament; as, we are both in the same category.

Webster Dictionary

Category (noun)
a collection of things sharing a common attribute;
"there are two classes of detergents"
Category (noun)
a general concept that marks divisions or coordinations in a conceptual scheme

Princeton's WordNet