A concrete category is a category
The object is called a generator of the category.
Often the term “concrete category” is used without implying the second condition of representability. This second condition however is important for the statement of concrete dualities induced by dual adjunctions?.
itself with the singleton set;
any category which has an object which is “free on one generator”, i.e which is the image under a functor left adjoint to of the singleton set. Then by definition of left adjoint and the above example, we have , naturally in . This abstract nonsense indicates the usual collection of examples of concrete categories: for instance monoids, groups, rings, algebras, etc. (… many more examples…)
Take to be the category of Banach spaces with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by (so for all in the source). Then there are two versions of that one may use: one where (for a Banach space) consists of every vector in , and one where consists of those vectors bounded by (so the closed unit ball in ). The first may seem more obvious at first, but only the second is representable (by a -dimensional Banach space).
“Only if” was proven in (Isbell). To prove it, note that if is a faithful functor, then it is injective on equivalence classes of regular subobjects. For suppose that is the equalizer of , and is the equalizer of . If as subobjects of , then since and so , we must also have ; hence (since is faithful) , so that factors through . Conversely, factors through , so we have as subobjects of . Since is regularly well-powered, it follows that any category admitting a faithful functor to must also be so.
(Actually, Isbell proved a more general condition which applies to categories that may lack finite limits.)