I started reading about categories in MacLane's book. This distinction between categories and metacategories is not clear to me.
Is there any example to enlighten their usage? For example, what about the metacategory of sets vs the category of sets, or the same with groups.
What Mac Lane calls a metacategory is any model of the axioms of category, whether these are sets are not. So, if the objects don't form a set, but a proper class, than you have a metacategory rather than a category. These things are discussed in the first several pages of the book. It's best not to get stuck with these details but rather just read it and if it gets too confusing, keep on reading and come back to it later. The thing to remember is that in a category we do not demand all morphisms to form a set. If each hom-set in the category is indeed a set, the category is called locally small. If the objects form a set, then the category is called small. These are size issues that are certainly important but can be confusing initially. Examples to keep in mind: all sets and all functions do not form a small category since the collection of all sets is a proper class, so this is a metacategory (or a large category). The category whose objects are the sets $\{1,2,3\},\{5,6,7,8\}$ with all functions between them is a small category. I hope this helps.
