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Concrete category - Definition

Concrete category - Definition: Encyclopedia II - Concrete category - 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 s ...

Concrete category: Encyclopedia II - Concrete category - Definition

Concrete category - 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.