My question is quite simple, I would like to know if we can define the category of the categories, unlike Cat which is the category of the small categories. By the way, are there any particular reason why we define Cat in this way?
Thanks in advance
2026-04-08 19:09:36.1775675376
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Is there a category of categories?
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In a certain sense, yes, there is, but it's not a category. There is a 2-category of "all" categories, given a choice of a Grothendieck universe. Have a look here: http://ncatlab.org/nlab/show/Cat
No, there is no category of all categories. Likewise, there is no "set of all sets" or "class of all classes" or..."mother of all mothers". Think about it: the mother of all mothers, if it existed, would be a mother (by definition), so she would have to be the mother of herself. Pretty akward.
Nevertheless, in mathematics it is important to be able to consider the collection of all categories (small and large) and the morphisms between them (functors). This collection is given the name CAT, but it is not a category itself for the same Russel's paradox type of argument given above. CAT is a quasicategory or a metacategory or something super... to indicate that it belongs to a higher hierarchy of structures. In "the joy of cats" book CAT is a called a quasicategory, but this name is now being used in another sense by most authors. I would encourage you to read the book, since it gives a good understanding of the size issues involved and an excellent intro to category theory.