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All the fun at flat module gave me the idea for the following quick test of one’s categorical level. In the variety of descriptions at flat module we have:
Which do you prefer?
If you answer 1, then your categorical level is 0.
If you answer 2, then your categorical level is 1.
If you answer 3, then your categorical level is at least 2.
What made me think of this (abject silliness) was that I was surprised to realise that I actually like the second description best, thus putting my categorical level higher than I normally think of it being.
Leaning more to 3 than 2, I think. It’s 3 which in my mind explains why we should care about 2.
Somewhere between 2 and 3, but closer to the former.
Both (2) and (3) make me happy. (1) just seems tedious. Sometimes I do think on that level, but I feel like I’m deliberately dumbing things down to do so. (To be sure, sometimes I’m dumbing things down for my own sake!)
Surely (3) is better than (2), but I’m not sure that I understand it yet in this case.
I feel that there are other levels than 1,2, and 3. None of these explain why the word ’flat’! I also think that the important idea is the intuition behind the relationships between the three levels. None is ‘Dumbing down’ to quote Toby. By the way you may knowthat there is a lovely silly book called something like ’mathematics made difficult’ which has the paragraph:
“This section is about addition. The fact that the reader has been told this does not necessarily mean that he knows what the section is about, at all. He still has to know what addition is, and that he may not yet know. It is the author’s fond hope that he may not even know it after he has read the whole section.”
Knowing what a ’flat module’ is really requires one to understand all three levels at a deep level!!!! :-)
I knew 3, and had seen and largely ignored 1 as unilluminating, but I squealed happily a little when I saw 2, because it's nice in itself as well as helping to make sense of 3. So I suppose I'd oscillate between 2 and 3. But I'd also agree with Tim that
the important idea is the intuition behind the relationships between the three levels.
There's an interesting collection of ideas involving flatness, but I don't quite know how they overlap: there's 'torsor' = 'flat presheaf', (a different sense of) 'torsor' = 'locally representable presheaf', and (for nice rings) 'flat module' = 'locally free module'. I wonder if there's a context in which 'locally foo' = 'filtered colimit of foos', or something similar, would make sense.
What’s funny here about the word ’torsor’ as a synonym for ’flat presheaf’ is its kinship to the word ’torsion’, as in ’torsion group’, which is sort of opposite to being flat (a -module is flat iff it is torsion-free). Various people have asked about the the origin of the term ’flat’ as applied to modules, but I’ve never heard any consensus on the matter. My own guess is that ’flat’ was originally proposed as a generalization of ’torsion-free’ (which is btw a very type 1 formulation), perhaps under the influence of the moon of Frenet-Serret frames and curves in , where a curve is ’flat’ (lies in a plane) iff its torsion vanishes.
Finn, I normally think of “projective” as a synonym for “locally free” (becomes free after localizing at a prime, i.e., after taking stalks at a point in the spectrum).
Does anyone have a contact who could give the origin of the use of ‘plat’ in this context. There is an article by Dieudonné, Adv.in Math., 3 (1969) 233-321, in which he discusses some of the terminology from a motivational point of view. I used to have a copy but do not now know where it is. (I do not have access to AIM as it is now owned by one of the *******<- expletive deleted!) I think there was a very geometric motivation for the term.
To return to the original question, surely the true measure of categorical level should be to appreciate the links between the concepts, as category theory is more about the relations between things than just the things themselves. ;-)
Various people have asked about the the origin of the term ’flat’ as applied to modules, but I’ve never heard any consensus on the matter.
I thought it came from algebraic geometry: a ring homomorphism is flat if the corresponding morphism of schemes varies in some flattish way.
some flattish way
Could you amplify on that, Mike?
Yes, but what I want is a bit more intuition about that ‘flattish way’ as you so neatly put it.
I vaguely remember that some intuition about flatness in algebraic geometry is discussed at some length in the Commutative algebra monograph by Eisenbud.
Perhaps if someone has a copy they could look at p.153 as the Google book version seems to indicate a link there with vanishing first derivatives, if handling rings of germs of functions. The previewed excerpt is too short to make much sense.
(I noted that near one of the entries on flatness was one on the Rees algebra of an ideal. Does anyone have information on this? I ask for personal reasons since David Rees was from my home town and was taught by my father in school.)
I don’t think the word “flat” appearing on p.153 has anything to do with flat modules. He is merely using the word to describe functions which have all derivatives vanishing. The whole next chapter is dedicated to flatness, but reading through it, I don’t feel particularly enlightened regarding the (geometric) intuition for flatness.
Thanks. The mystery remains therefore! (for the moment)
but see this perhaps.
(Added later: The answers there are very good, and perhaps some digest of that should be put somewhere on the Lab.)
Andrew, thanks for the original question. :-)
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