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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Wed, Nov 11, 2009 at 10:36 PM, Andrew Salch <asalch@math.jhu.edu> 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?
Suppose the class I is described by a first-order formula a(x) and the classes C_i are described by a first-order formula c(x,i), i.e: I = {i | a(i)} C_i = {x | c(x,i)} Then the intersection of the C_i's is a class because it is described as D = {x | forall i, a(i) => c(x,i)} Your condition that the C_i's are contained in a class C is redundant because we may always take C=V, the universe. With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Andrew Salch wrote:
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?
In ZF, "proper class" means "first-order formula," the formula phi(x) being regarded as a stand-in for "the class of sets x such that phi(x)". In this case the only meaning that can be given to "a family of classes indexed by a class" is that we have a formula with two variables phi(x,y), so that each y indexes "the class of sets x such that phi(x,y)". In this case, as Andrej has said, the intersection of these classes is represented by the formula "for all y, phi(x,y)", so there is no problem. In set-class theories such as NBG and MK, classes are things in their own right (rather than having to be represented by first-order formulas). However, in neither of these theories can a class be an *element* of another class, so "a family of classes indexed by a class" can't mean "a class whose elements are classes" or "a class function whose values are classes"---instead it has to be similar to the meaning in ZF, such as a single class P whose elements are ordered pairs, where each set y indexes the class { x | (x,y) \in P }. Again in these theories there is no trouble with the intersection of such an indexed family of classes, it is just the class { x | (x,y) \in P for all y }. However, in your example:
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
I don't think that "the class of reflective topologizing subcategories of X" is a legitimate class in the sense of ZF, NBG, or MK. Each reflective topologizing subcategory is itself a proper class, and (in NBG and MK) classes can't be elements of other classes, so there is no class of all reflective topologizing subcategories of X, and thus no class to "index" the intersection you want to take. I think you *can* still define this intersection in MK, however: it is the class { a \in ob(X) | a \in W for any reflective topologizing subcategory W of X that contains both Y and Z } This is allowable in MK because its comprehension axiom for classes allows formulas that can quantify over classes. The comprehension axiom of NBG only allows formulas that do not quantify over classes, so I don't think this construction can be done in NBG. Note that while NBG is conservative over ZFC (it can't prove any new theorems about sets), MK is not. All of these problems go away, of course, if you assume a Grothendieck universe U and replace your sets by U-small sets and your classes by U-large ones.
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.
I'm not sure if this is quite what you're looking for, but my paper "Set theory for category theory" addresses some of these issues (arXiv:0810.1279). Best, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Andrew Salch wrote:
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?
Mike Shulman's answer is more comprhensive than what I could write, but I want to stress two points. For the first point, I quote Mike:
I think you *can* still define this intersection in MK, however
So there is a well-known theory of sets and classes in which you can do this. And the second point is that, unless you are particulary interested in the foundational and logical issues of size and categories, the first point is the only thing that you should care about. (Mike and I are interested in those issues, which is why he wrote a detailed answer and I was happy to read it. But "working" category theorists shouln't have to be.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
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Andrej Bauer -
Andrew Salch -
Eduardo J. Dubuc -
Michael Shulman -
Toby Bartels