Concrete Category

A concrete category is a pair (C,U) such that

• C is a category, and
• U is a faithful functor C → Set (the category of sets and functions).

The functor U is to be thought of as a forgetful functor, which assigns to every object of C its "underlying set", and to every morphism in C its "underlying function".

A category C is concretizable if there exists a concrete category (C,U); i.e., if there exists a faithful functor U:C → Set. All small categories are concretizable: define U so that its object part maps each object b of C to the set of all morphisms of C whose codomain is b (i.e. all morphisms of the form f: a → b for any object a of C), and its morphism part maps each morphism g: b → c of C to the function U(g): U(b) → U(c) which maps each member f: a → b of U(b) to the composition gf: a → c, a member of U(c). (Item 6 under Further examples expresses the same U in less elementary language via presheaves.) The Counter-examples section exhibits two large categories that are not concretizable.