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 protypical conctete 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.
If C is a concrete category, then there exists a forgetful functor F : C → Set which assigns to every object of C the underlying set and to every morphism in C the corresponding function. This functor is faithful, i.e. it maps different morphisms between the same objects to different functions (it may however map different objects to the same set). In the formal approach, a concrete category is defined as a category together with a faithful functor into the category of sets.
The category Toph where the objects are topological spaces and the morphisms are homotopy classes of continuous functions is an example of a category that is not concrete. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions.
