 Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structurepreserving 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 morphisms, 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 morphism from A to A, and the composition of a morphism from A to B followed by a morphism from B to C must be a morphism from A to C.^{[1]}
Contents
Definition
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 Counterexamples section exhibits two large categories that are not concretizable.
Remarks
It is important to note that, contrary to intuition, concreteness is not a property which 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.
The requirement that U be faithful means that it maps different morphisms between the same objects to different functions. However, U may map different objects to the same set and, if this occurs, it will also map different morphisms to the same function.
For example, if S and T are two different topologies on the same set X, then (X,S) and (X,T) are distinct objects in the category Top of topological spaces and continuous maps, but mapped to the same set X by the forgetful functor Top → Set. Moreover, the identity morphism (X,S) → (X,S) and the identity morphism (X,T) → (X,T) are considered distinct morphisms in Top, but they have the same underlying function, namely the identity function on X.
Similarly, any set with 4 elements can be given two nonisomorphic group structures: one isomorphic to ; the other isomorphic to .
Further examples
 Any group G may be regarded as an "abstract" category with one object, , and one morphism for each element of the group. This would not be counted as concrete according to the intuitive notion described at the top of this article. But every faithful Gset (equivalently, every representation of G as a group of permutations) determines a faithful functor G → Set. Since every group acts faithfully on itself, G can be made into a concrete category in at least one way.
 Similarly, any poset P may be regarded as an abstract category with a unique arrow x → y whenever x ≤ y. This can be made concrete by defining a functor D : P → Set which maps each object x to and each arrow x → y to the inclusion map .
 The category Rel whose objects are sets and whose morphisms are relations can be made concrete by taking U to map each set X to its power set 2^{X} and each relation to the function defined by . Noting that power sets are complete lattices under inclusion, those functions between them arising from some relation R in this way are exactly the supremumpreserving maps. Hence Rel is equivalent to a full subcategory of the category Sup of complete lattices and their suppreserving maps. Conversely, starting from this equivalence we can recover U as the composite Rel → Sup → Set of the forgetful functor for Sup with this embedding of Rel in Sup.
 The category Set^{op} can be embedded into Rel by representing each set as itself and each function f: X → Y as the relation from Y to X formed as the set of pairs (f(x),x) for all x ∈ X; hence Set^{op} is concretizable. The forgetful functor which arises in this way is the contravariant powerset functor Set^{op} → Set.
 It follows from the previous example that the opposite of any concretizable category C is again concretizable, since if U is a faithful functor C → Set then C^{op} may be equipped with the composite C^{op} → Set^{op} → Set.
 If C is any small category, then there exists a faithful functor P : Set^{Cop} → Set which maps a presheaf X to the coproduct . By composing this with the Yoneda embedding Y:C → Set^{Cop} one obtains a faithful functor C → Set.
 For technical reasons, the category Ban_{1} of Banach spaces and linear contractions is often equipped not with the "obvious" forgetful functor but the functor U_{1} : Ban_{1} → Set which maps a Banach space to its (closed) unit ball.
Counterexamples
The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not exist any faithful functor from hTop to Set was first proven by Peter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalenceclasses of functors" also fails to be concretizable.
Implicit structure of concrete categories
Given a concrete category (C,U) and a cardinal number N, let U^{N} be the functor C → Set determined by U^{N}(c) = (U(c))^{N}. Then a subfunctor of U^{N} is called an Nary predicate and a natural transformation U^{N} → U an Nary operation.
The class of all Nary predicates and Nary operations of a concrete category (C,U), with N ranging over the class of all cardinal numbers, forms a large signature. The category of models for this signature then contains a full subcategory which is equivalent to C.
Relative concreteness
In some parts of category theory, most notably topos theory, it is common to replace the category Set with a different category X, often called a base category. For this reason, it makes sense to call a pair (C,U) where C is a category and U a faithful functor C → X a concrete category over X. For example, it may be useful to think of the models of a theory with N sorts as forming a concrete category over Set^{N}.
In this context, a concrete category over Set is sometimes called a construct.
Notes
 ^ Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), AMS Chelsea, ISBN 9780821816462
References
 Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0471609226. (now free online edition).
 Freyd, Peter; (1970). Homotopy is not concrete. Originally published in: The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168. Republished in a free online journal: Reprints in Theory and Applications of Categories, No. 6 (2004), with the permission of SpringerVerlag.
 Rosický, Jiří; (1981). Concrete categories and infinitary languages. Journal of Pure and Applied Algebra, Volume 22, Issue 3.
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