Concrete category
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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.
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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, 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 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 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). Abstract and Concrete Categories (http://www.math.uni-bremen.de/~dmb/acc.pdf). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)