intersections of classes

[Andrew Salch <asalch@math.jhu.edu>]

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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