Жиынға адал functor жабдықталған санаттар

Category equipped with a faithful functor to the category of sets
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, see Relative concreteness below). 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.

Any group G may be regarded as an "abstract" category with one arbitrary 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 G set (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 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 supremum preserving maps. Hence Rel is equivalent to a full subcategory of the category Sup of complete lattices and their sup preserving 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 Setop 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 Setop is concretizable. The forgetful functor which arises in this way is the contravariant powerset functor Setop → 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 Cop may be equipped with the composite Cop → Setop → Set. If C is any small category, then there exists a faithful functor P : SetCop → Set which maps a presheaf X to the coproduct By composing this with the Yoneda embedding Y:C → SetCop one obtains a faithful functor C → Set. For technical reasons, the category Ban1 of Banach spaces and linear contractions is often equipped not with the "obvious" forgetful functor but the functor U1 : Ban1 → Set which maps a Banach space to its (closed) unit ball. The category Cat whose objects are small categories and whose morphisms are functors can be made concrete by sending each category C to the set containing its objects and morphisms. Functors can be simply viewed as functions acting on the objects and morphisms.

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 equivalence classes of functors" also fails to be concretizable.

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 SetN. In this context, a concrete category over Set is sometimes called a construct.