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.

The traditional way of describing the similarities and differences between constituents is to say that they belong to categories of various types. Thus, words like boy, girl, man, woman, etc. are traditionally said to belong to the category of Nouns, whereas words like a, the, this, and that are traditionally said to belong to the category of Determiners.

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.

The use of the word ‘factor’ shows the explicit intention of categorists to work with functions in an algebraic manner: a category is an algebra of functions.

Category

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