Concrete Category
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A Concrete Category is a category that ...
- See: Class (Set Theory), Category (Category Theory), Faithful Functor, Category of Sets, Mathematical Structure, Morphism, Category of Topological Spaces, Category of Groups, Homotopy Category of Topological Spaces.
References
2017
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Concrete_category Retrieved:2017-6-4.
- 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 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 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.
- 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 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.