There should not be any problem taking intersections as well as unions of large classes indexed by large classes. Problems arise with products or coproducts in categories, but not if the categories are posets. Andrew Salch wrote:
I have a question for the category theorists which is unfortunately just an issue about sets and classes that I hope some of you have thought about before, and can help me with: let C be a class, and consider a family of subclasses C_i of C, which are indexed by an index class I. Am I allowed to take the intersection of a family of classes indexed by a class? Is the result a class?
What I am really thinking of, here, is the situation that C is the class of objects in an abelian category X; I have two reflective topologizing subcategories Y,Z of X; and I would like to know that there exists a smallest reflective topologizing subcategory of X containing both Y and Z. The intersection of reflective topologizing subcategories is again reflective and topologizing, so I would like to be able to take the intersection of all the reflective topologizing subcategories of X containing both Y and Z (or, what comes to the same thing since all these subcategories are full subcategories, the full subcategory generated by the intersection of the object classes of all the reflective topologizing subcategories of X containing both Y and Z). However this is an intersection of classes, indexed by a class, and in general one can't expect any of these classes to be sets.
When the abelian category X is the category of modules over a commutative ring, then the class of reflective topologizing subcategories of X forms a set, so one can take this intersection without any problems; but I do not suspect that this will be true for all abelian categories.
More generally, if there is a book or paper on set theory which covers some of the basic operations you can and can't do with classes, "for the working mathematician," I'd really like to hear about it.
Thanks, Andrew S.
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