In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets.

A concrete category, when defined without reference to the notion of a category, consists of a class of objects, each equipped with an underlying set; and for any two objects A and B a set of functions, called homomorphisms, from the underlying set of A to the underlying set of B. Furthermore, for every object A, the identity function on the underlying set of A must be a homomorphism from A to A, and the composition of a homomorphism from A to B followed by a homomorphism from B to C must be a homomorphism from A to C.

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

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".

It is customary to call the morphisms in a concrete category homomorphisms (e.g., group homomorphisms, ring homomorphisms, etc.) Because of the faithfulness of the functor U, the homomorphisms of a concrete category may be formally identified with their underlying functions (i.e., their images under U); the homomorphisms then regain the usual interpretation as "structure-preserving" functions.

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.

Contrary to intuition, concreteness is not a property that a category may or may not satisfy, but rather a structure with which a category may or may not be equipped. In particular, a category C may admit several faithful functors into Set. Hence there may be several concrete categories (C, U) all corresponding to the same category C.

In practice, however, the choice of faithful functor is often clear and in this case we simply speak of the "concrete category C". For example, "the concrete category Set" means the pair (Set, I) where I denotes the identity functor Set → Set.