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In mathematics, a 'concrete category' is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. The prototypical concrete category is 'Set', the category of sets and functions.
Most categories considered in everyday life are concrete; examples are 'Top', the category of topological spaces and continuous functions, and 'Grp' the category of groups and group homomorphisms.
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 'Grp' of groups, any set with 4 elements can be given two non-isomorphic group structures, (namely, or ), but to check if two group homomorphisms between groups ''G'' and ''H'' are equal, we need only check that the underlying set functions are equal.
A category ''C'' is ''concretizable'' if there exists a faithful functor from ''C'' into 'Set'.
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 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 attempt to define such a functor.
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''.
★ Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). ''Abstract and Concrete Categories'' (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
Most categories considered in everyday life are concrete; examples are 'Top', the category of topological spaces and continuous functions, and 'Grp' the category of groups and group homomorphisms.
| Contents |
| Definition |
| Not all categories are concrete |
| Alternate definition |
| References |
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 'Grp' of groups, any set with 4 elements can be given two non-isomorphic group structures, (namely, or ), but to check if two group homomorphisms between groups ''G'' and ''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 '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 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 attempt 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). ''Abstract and Concrete Categories'' (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
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