intersections of classes

intersections of classes

[Andrew Salch]

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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Re: intersections of classes

[Andrej Bauer]

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


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Re: intersections of classes

[Eduardo J. Dubuc]

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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Re: intersections of classes

[Michael Shulman]

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


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Re: intersections of classes

[Toby Bartels]

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


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