8
$\begingroup$

When a group is defined, we ask that the elements belong to a set. If we allow them to belong to a proper Class we get a GROUP.

What is the advantage of working with groups? What properties do we lose when we work with GROUPS?

An example of the notation can be found here: http://www.mathe2.uni-bayreuth.de/stoll/papers/games12.pdf

  • 3
    I would say that is a matter of historical development, in the sense that groups precede classes. And one disanvantage that comes to my head now, how do you define the quotient GROUP? (Since partitions of classes require to define something that it is not a class.)2011-11-18
  • 0
    If I had to venture a guess, the issue is that one wants the axiom of comprehension to construct certain kinds of subgroups/quotients. However, you can define group objects in any monoidal category, so we don't necessarily need our elements to belong to a set. Perhaps a better question is, what benefits are there to having groups whose elements belong to a proper class with no additional structure? Even without there being any benefits to working with sets, if there aren't benefits to working with proper classes, why put people in unfamiliar territory?2011-11-18
  • 0
    @Iasafro, Perhaps as the range of a homomorphism. But it would probably be difficult to prove that every normal SUBGROUP is the kernel of a homomorphism. On the other hand, if the normal subgroup is small enough to be a set, then so are all the cosets.2011-11-18
  • 5
    People have found ample occupation when dealing with plain all small groups... What exactly do you want to achieve by considering GROUPS? What do you expect would be different in, say, your favorite textbook on group theory?2011-11-18
  • 2
    @Mariano: Something pointed out to me today: let $V$ be the universe of all sets, then the permutation ‘group’ $\textrm{Sym}(V)$ in fact contains _every_ (small) group as a subgroup.2011-11-18
  • 0
    @Mariano Most/all the theory is done on groups, I want to know if there is a real Mathematical reason for this. I don't know much of this topic so I have no intuition how bad things behave when one works with proper classes. I'm specially interested in subtle hinderings one has to beware of.2011-11-18
  • 2
    @ZhenLin, but I would imagine it does not contain all *GROUPS* as subGROUPS so we did not get much!2011-11-19
  • 4
    In typical theories of (sets and) classes, all the PERMUTATIONS of $V$ won't even form a GROUP, because the individual PERMUTATIONS are proper classes and thus can't be members of a class. If we enlarge our world more, to allow collections of proper classes, then Sym(V) is such a super-class. It contains all class-sized groups (by Cayley) but not all super-class sized ones.2012-11-11
  • 6
    Whatever set- or class-theoretic level you allow for groups, you'll soon want higher levels, for (the natural construction of) quotient groups, automorphism groups, etc. So either work with sets, so that htese higher levels are available, or, if you really want proper classes, work in a theory that also allows super-classes, super-duper-classes, etc. I find it simplest to work in set theory.2012-11-11
  • 0
    One reason that hasn't been mentioned yet: because virtually all of the group-like entities we care about are small! Keep in mind that groups are often studied for or developed from their actions on other entities; for instance, the groups $GL_n(\mathbb{R})$, the permutation groups and subgroups of them, etc. These entities turn out to have an astonishingly interesting theory in their own right and a lot of commonality to them, but don't forget that they primarily arise from other existing 'small' structures.2014-02-05
  • 0
    A slightly cynical view might be the following: Group theorists are group theorists, not category theorists. You get group theorists who are also category theorists, but some are geometers instead. Working with GROUPS would mean their foundation would be shaken and they would have to learn a lot again. As it is, some of them probably like looking at groups as GROUPS. Others don't know the difference between groups and GROUPS. It is like trying to get everyone in Europe to speak Esperanto (which, incidentally, I speak like a native) - perhaps a good idea, but not really workable...2014-02-05
  • 0
    @user1729 What does it mean to speak Esperanto like a native?2014-02-05
  • 0
    @Michael It means being unable to speak it...2014-02-06

1 Answers 1

4

For the same reasons why most of the time we only talk about sets and not proper classes in general. In vast majority of contexts, proper classes are not needed at all.

In practice, we're interested in small objects. Proper classes arise mostly when we want some kind of very universal object. But even then the universality is usually rather superficial – you can just think that the supposed proper class is actually a set in a proper extension of the universe, and its only universal for objects of small size. In that way, being a proper class is pretty relative.

A different problem is that for proper classes we don't have many constructions and theorems that are commonly used, like equivalence relations, quotients (so no isomorphism theorems!, no presentations!), wellorderings. If we want to add those things we're basically doing the same as what I mentioned in the previous paragraph: we make the universe larger.

I'm simplifying a bit, there are some subtleties involved, and many of these obstacles can be circumvented using some clever tricks, but generally proper classes are not important, nor do they happen too often if you aren't actually looking for them. If you want really big objects, you'd be better off assuming there's a sufficiently large strongly inaccessible cardinal and working in a restricted universe. That makes for far less of an ontological headache, and doesn't really make you lose anything of interest, as far as I can tell.