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I thought I would take the plunge. Here is something from the Wikipedia page on mduli space:
the term “modulus” is used synonymously with “parameter”
That deos not accord well with other uses of ’modulus’ nor with its Latin origins, but although I do not know much about moduli spaces, I did think of them as parametrising families of ’things’. Somehow however ’modulating’, probably because of the other English senses of the word, fails to focus in on this parametrising aspect. I am not convinced it is a good word for the idea but cannot come uip with a better one!
There is also the link of moduli spaces / stacks with deformation theory so ’small’ variations of the map lead to ’small’ in the structure of the objects being ’classified’.
Search Ben-Zvi’s review for occurrences of “modulate”.
Urs, thanks for that link. I will read his ’review’. I like his use of modulate' as a way of sayingvary’. What I wonder is whether ’modulating map’ is an accurate description of what you describe, jsut because it is not the map that is varying. Perhaps it is the use of the present participle that seems strange ’modulation map’ might be better.
The “classifying map” is also not that which is classified, but that which classifies.
… but as I said, the map is not modulating. (I think this is ude in part to some quirks of English. It is a very awkward language.) In the other case the map that does the classifying in some sense.
The modulating map is modulating (by sending each point in its domain to its the corresponding modulus) in just the same sense as the classifying map is classifying (by sending each point to its corresponding class).
I am fine with Mike’s argument that from some perspective all modulation is itself classification (that’s why we say object classifier and not “moduli stack of objects”). And I have understanding for people not wanting to follow terminology as in Ben-Zvi’s article, because it is not widely established. But your argument why there is some problem of principle I don’t follow.
The modulating map is modulating in just the same sense as the classifying map is classifying. That’s the whole point of the terminology.
I am fine with Mike’s argument that from some perspective all modulation is itself classification (that’s, after all, why we say “object classifier” and not “moduli stack of objects”). And I have understanding for people not wanting to follow terminology as in Ben-Zvi’s article, because it is not widely established. But your argument that there should be a problem of principle I don’t follow.
You say:
The modulating map is modulating in just the same sense as the classifying map is classifying.
It is not
As I understand the use of the word ’modulating’ in English to be a synonym of varying' andvarying map’ does not give the meaning you intend, I disagree. … but it does not matter very much. It is certainly not a ’problem of principle’. The problem is probably that I come with a baggage of usage in everyday language in the UK that is not the same for everyone. The sort of thing I mean is FM radio and the M stood for ’modulation’ thus ’variation’. If I applied my synonym to the quote above it would start by saying. ’The modulating map is varying’, and I fail to see that in any usual sense of ’varying’. However this is not very useful in pushing the theory of such things further, so ….
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