nLab CAT

Recall that Cat is generally used to denote the (or a) category (or 2-category) of categories (and functors and natural transformations). Of course, we cannot usually expect to have a category of all categories, for the same reasons we cannot have a set of all sets, so $Cat$ usually denotes the category of small categories, whatever that means in your chosen foundations. Of course, $Cat$ is then not a small category but a large category.

In some foundations, such as Grothendieck universes, it nevertheless makes sense to talk about the “category of large categories,” which is itself not a large category (for the same reason) but a “very large” category (some older work calls it a metacategory). It is fairly common to denote this “very large” category (or 2-category) by $CAT$.

References

• Jiri Adamek, Horst Herrlich, and George Strecker, Abstract and concrete categories: the joy of cats. free online

category: category