9

Why is it necessary to define germs of functions (in my case, for foliations, but my question is in general)? does any inconsistency arises if instead of using a germ in some context, I use representative element of the germ?

Marra
  • 5,039
  • I guess because you reduce your work since you're seeing a lot different elements of a larger space as a single element in the quotient. That's just a hint, what you think? – PtF May 31 '15 at 20:40

1 Answers1

5

It's a matter of convenience (great convenience). You can ask the same question about factor groups (or factor anything). Why work with the equivalence class $[g]$ and not just with a representative from that class. Well, if you want to form a quotient group then it is a lot more convenient to consider the elements of the quotient to be equivalence classes of elements rather than make an arbitrary choice for a representative from each class (try it if you're in doubt).

This is a general phenomenon: If you make arbitrary choices, they'll come back to haunt you. If, somehow, you can make a canonical choice of (of a representative from each equivalence class) then you're fine (usually). But if no such natural choice exists (or is used for a particular choice) then it is almost guaranteed to lead to a lot of mess.

For an extreme example, you might say that all of set theory should be reduced to the study of a single representative of each cardinality. After all, a set is completely determined by its cardinality, so would it not be simpler to chuck away all sets and just choose (arbitrarily!!) a single set of each cardinality? Less sets to study, hence easier, right? Well, not quite. Suppose this is done and you now want to describe addition: $+:\mathbb N \times \mathbb N \to \mathbb N$. Oops, little problem here, both domain and codomain are countable, so in our world they are now one and the same set. Unpleasant.

Ittay Weiss
  • 81,796
  • I can understand that. But it seems to me, from what you told, that working with germs (or a class) to solve a local problem (local as in topology) is using a tool that avoids problems which may arise globally... Is that correct? – Marra Apr 30 '13 at 23:47
  • No, this has nothing to do with local and global issues. If you wish you can forget the term germ ever existed and work with representatives. Your arguments then will be longer and more cumbersome due to the lack of canonical choice of representative. – Ittay Weiss May 01 '13 at 00:07
  • 1
    I don't want to ignore them, I want to understand why they are useful... – Marra May 01 '13 at 00:17
  • I explained why. They are convenient since they make proofs shorter and less cumbersome. – Ittay Weiss May 01 '13 at 00:23
  • I still don't see why. Can you show me an example where the option to not use a germ instead of a representative makes things easier? – Marra May 01 '13 at 00:48
  • 2
    I agree with @Marra that from your answer is not really clear why germs are useful. I was wondering about the same question of the OP since, from my limited perspective, the only difference made by the use of germs are those “it's easily checked that the definition doesn't depend on the representative” etc.etc. A concrete example may be useful. – pppqqq Aug 22 '14 at 19:10
  • I've come to realize that it works easier when working with holonomy groups (saving a lot of work by not having to deal with open subsets for a 'chain' of functions...) and also works great with sheaf theory... – Marra Aug 23 '14 at 22:13
  • @IttayWeiss, I agree with Marra and pppqqq, I do not understand your writing either. – Mikkel Rev Feb 02 '19 at 19:20