Why are germs of functions important?

Question

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?

Answer 1

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.