> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/10-categories/concrete-category.md). # Concrete category A **concrete category** is a category equipped with a *faithful functor* to 𝗦𝗲𝘁 category. This functor makes it possible to think of the objects of the category as *sets with additional structure*, and of its morphisms as *structure-preserving functions*. Many important categories have obvious interpretations as concrete categories; for example, the category of topological spaces and the category of groups, and, trivially, also 𝗦𝗲𝘁 category itself. On the other hand, the homotopy category of topological spaces is *non-concretizable* i.e. it does not admit a faithful functor to the category of sets. A concrete category, when defined without reference to the notion of a category, consists of a class of objects, each equipped with an underlying set; and for any two objects A and B a set of functions, called morphisms, from the underlying set of A to the underlying set of B. Furthermore, for every object A, the identity function on the underlying set of A must be a morphism from A to A, and the composition of a morphism from A to B followed by a morphism from B to C must be a morphism from A to C. A **relatively concrete category** is a category that is equipped with a *faithful functor* to another category.