Concrete Category and Abstract Category

From Analysis and Its Foundations By Eric Schechter:

A concrete category consists of a collection of objects and a collection of morphisms.

I am curious what an "abstract category" is?

I found that the Wikipedia page for category theory seems to be only about concrete categories.

Thanks and regards!

1 Answer

This is just an incorrect use of the term "concrete category." Informally, a concrete category is a "category of sets and functions" - that is, the objects are certain sets, and the morphisms are certain functions between them. More formally, a concrete category is a category $C$ equipped with a faithful functor $C \to \text{Set}$, often called the "underlying set" functor or the "forgetful" functor.

Most of the first examples of categories people meet - sets, groups, rings, modules - are concrete, because they have objects given by sets with extra structure and morphisms given by maps respecting that structure. To get examples of "abstract" categories, just meaning categories that aren't obviously equipped with faithful functors to $\text{Set}$, you can take opposite categories; already it's not obvious from first principles how to think of $\text{Set}^{op}$ as a concrete category (although it can be done).