category

noun2

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.