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Bestselling Series. Harry Potter. Popular Features. 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 ". 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. 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. The class of all N -ary predicates and N -ary operations of a concrete category C , U , with N ranging over the class of all cardinal numbers, forms a large signature.
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The category of models for this signature then contains a full subcategory which is equivalent to C. 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 example, it may be useful to think of the models of a theory with N sorts as forming a concrete category over Set N.
From Wikipedia, the free encyclopedia. Categories : Category theory.