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?
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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. |
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