
In mathematics, a concrete category is a category (category theory) in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are function (mathematics)s between those sets, and the composition of morphisms is the composition of functions. The prototypical concrete category is Category of sets, the category of sets and functions. Most categories considered in everyday life are concrete; examples are Category of topological spaces, the category of topological spaces and continuous functions, and Category of groups the category of group (mathematics) and group homomorphisms. == Definition == A concrete category is formally defined as follows: *a category ''C'' *a faithful functor ''F'' : ''C'' → Set The faithful functor ''F'' is typically thought of as a forgetful functor, which assigns to every object of ''C'' its "underlying set", and to every morphism in ''C'' the corresponding function. Thus, a concrete category ''C'' consists not just of ''C'' itself, but of the category ''C'' and a corresponding forgetful functor ''F''. In practice, the forgetful functor is usually clear, and we simply speak of the "concrete category ''C''". The requirement that ''F'' be faithful means that different morphisms between the same objects map to different functions. (However, different objects may map to the same set, and morphisms between different objects may map to the same function.) For example, in the concrete category category of groups of groups, any set with 4 elements can be given two non-isomorphic group structures, (namely, Z/2 × Z/2 or Z/4), but to check if two group homomorphisms between groups ''G'' to ''H'' are equal, we need only check that the underlying set functions are equal. == Not all categories are concrete == A category ''C'' is ''concretizable'' if there exists a faithful functor from ''C'' into Set. The category homotopy category of topological spaces, where the objects are topological spaces and the morphisms are homotopy 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 statement that hTop is not concretizable says more than this simple observation, however; it asserts that there does not exist ''any'' faithful functor from hTop to Set, no matter how we choose to define such a functor. == Alternate definition == Some authors use a more general definition of concrete category, where an arbitrary category ''X'', (sometimes called the ''base category'') takes the place of Set. In this case, we say that a ''concrete category over X'' consists of a category ''C'' and a faithful functor ''F'' : ''C'' → ''X''. In this case, a concrete category over Set is sometimes called a ''construct''. == References == * Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). [http://www.math.uni-bremen.de/~dmb/acc.pdf ''Abstract and Concrete Categories'']. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition) category theory
Are there any categories which do not allow a faithful functor into Set? : Sure, tons of them. But are they "natural"? User:Phys 22:18, 15 Nov 2003 (UTC) ::Ask an algebraic topologist. User:Revolver 10:08, 27 Apr 2005 (UTC)
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Words begining with Concrete_category:
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Concrete category
In mathematics, a concrete category is a category (category theory) in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are function (mathematics)s between those sets, and the composition of morphisms is the composition of functions. The prototypical concrete category is Category of sets, the category of sets and functions. Most categories considered in everyday life are concrete; examples are Category of topological spaces, the category of topological spaces and continuous functions, and Category of groups the category of group (mathematics) and group homomorphisms. == Definition == A concrete category is formally defined as follows: *a category ''C'' *a faithful functor ''F'' : ''C'' → Set The faithful functor ''F'' is typically thought of as a forgetful functor, which assigns to every object of ''C'' its "underlying set", and to every morphism in ''C'' the corresponding function. Thus, a concrete category ''C'' consists not just of ''C'' itself, but of the category ''C'' and a corresponding forgetful functor ''F''. In practice, the forgetful functor is usually clear, and we simply speak of the "concrete category ''C''". The requirement that ''F'' be faithful means that different morphisms between the same objects map to different functions. (However, different objects may map to the same set, and morphisms between different objects may map to the same function.) For example, in the concrete category category of groups of groups, any set with 4 elements can be given two non-isomorphic group structures, (namely, Z/2 × Z/2 or Z/4), but to check if two group homomorphisms between groups ''G'' to ''H'' are equal, we need only check that the underlying set functions are equal. == Not all categories are concrete == A category ''C'' is ''concretizable'' if there exists a faithful functor from ''C'' into Set. The category homotopy category of topological spaces, where the objects are topological spaces and the morphisms are homotopy 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 statement that hTop is not concretizable says more than this simple observation, however; it asserts that there does not exist ''any'' faithful functor from hTop to Set, no matter how we choose to define such a functor. == Alternate definition == Some authors use a more general definition of concrete category, where an arbitrary category ''X'', (sometimes called the ''base category'') takes the place of Set. In this case, we say that a ''concrete category over X'' consists of a category ''C'' and a faithful functor ''F'' : ''C'' → ''X''. In this case, a concrete category over Set is sometimes called a ''construct''. == References == * Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). [http://www.math.uni-bremen.de/~dmb/acc.pdf ''Abstract and Concrete Categories'']. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition) category theory
Concrete category
Are there any categories which do not allow a faithful functor into Set? : Sure, tons of them. But are they "natural"? User:Phys 22:18, 15 Nov 2003 (UTC) ::Ask an algebraic topologist. User:Revolver 10:08, 27 Apr 2005 (UTC)
See other meanings of words starting from letter:
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CA | CB | CD | CE | CF | CG | CH | CI | CJ | CK | CL | CM | CN | CO | CP | CR | CS | CT | CU | CW | CX | CY | CZ |Words begining with Concrete_category:
Concrete_category
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