Concrete category

Summary

A concrete category is a category $\mathcal{C}$ together with a faithful functor $U$ from $\mathcal{C}$ to the category of sets. The functor $U$ assigns an underlying set to each object, and morphisms are determined by their underlying functions. Groups, rings, vector spaces and topological spaces with their usual forgetful functors are concrete categories. Not every category admits such a functor, so concreteness is a real condition on large categories.