Remarks
It is important to note that, contrary to intuition, concreteness is not a property which a category may or may not satisfy, but rather a structure with which a category may or may not be equipped. 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". For example, "the concrete category Set" means the pair (Set,I) where I denotes the identity functor Set → Set.
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.
For example, if S and T are two different topologies on the same set X, then (X,S) and (X,T) are distinct objects in the category Top of topological spaces and continuous maps, but mapped to the same set X by the forgetful functor Top → Set. Moreover, the identity morphism (X,S) → (X,S) and the identity morphism (X,T) → (X,T) are considered distinct morphisms in Top, but they have the same underlying function, namely the identity function on X.
Similarly, any set with 4 elements can be given two non-isomorphic group structures: one isomorphic to ; the other isomorphic to .
Read more about this topic: Concrete Category
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