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added more references:
[ operad structured due to Getzler-Jones… ]
and alternatively in
Maxim Kontsevich, around Def. 12 of Operads and Motives in Deformation Quantization, Lett.Math.Phys.48:35-72,1999 (arXiv:math/9904055)
Maxim Kontsevich, section 5.1 of Deformation quantization of Poisson manifolds, I, Lett.Math.Phys.66:157-216,2003 (arXiv:q-alg/9709040)
which was corrected in
and developed in detail in
which also shows the equivalence to Fulton-MacPherson 94.
I welcome this article (that’s an understatement), but are you sure about
unrelated to actual infinitesimal neighbourhoods in synthetic differential geometry.
I was under the impression that the sense of infinitesimal would be along lines long considered standard in algebraic geometry in terms of blowings-up, and that it wouldn’t be hard making this precise in SDG.
Okay, I don’t know of the relation then.
What I meant to say in that line is that there is no actual synthetic infinitesimals involved here, as in: no formal duals to nilpotent rings. But I gather you are saying that one may start with an alternative formulation where there are SDG-style infinitesimals, then apply blowups and arrive at the purely topological space that Fulton-MacPherson considered? Sounds interesting.
Oh, probably the SDG is hidden in the discussions of how the FM-compactifications are (semi-)algebraic manifolds? Where would the perspective of infinitesimals and blowups here be made explicit?
added pointer to
Scott Axelrod, Isadore Singer, Chern-Simons Perturbation Theory, in S. Catto, A. Rocha (eds.) Proc. XXthe DGM Conf. World Scientific Singapore, 1992, 3-45; (arXiv:hep-th/9110056)
Scott Axelrod, Isadore Singer, Chern–Simons Perturbation Theory II, J. Diff. Geom. 39 (1994) 173-213 (arXiv:hep-th/9304087)
who really considered the compactification first, and slightly expanded the Idea-section to reflect this. Also mentioned in the Idea section the quick informal description of the ASFM-compactification as the blowup of the fat diagonal.
If the desired space is being denoted $M[V]$ in Axelrod-Singer (for example on page 21 of 41), I think in Chern-Simons Perturbation Theory II, then they describe this compactification as a certain manifold with corners. I’d think a mild adaptation of SDG theory would be able to accommodate the appropriate constructions on manifolds-with-corners, so it still feels as though it’s pretty stark to explicitly deny anything more than a heuristic connection between the two senses of the word “infinitesimal”, as our article does. If I’m not misreading, their article does seem to suggest that the desired construction can be enacted in a category of manifolds-with-corners.
Interesting to see you amplify the physics-style intuition over the bare math facts this way. Please feel free to reword the entry as you see the need! All I tried to point out is the obvious: that the FM-compactification is a plain topological space, not an object in a smooth topos modelling SDG, so that, on the face of it there are no actual infinitesimals here. Where people speak of infinitesimal distance in FM, there is instead, by definition, ratios going plainly to actual zero in a topological space locally modeled on ordinary topological closed intervals. This can’t be controversial, since it is straight away the explicit definition.
Now I don’t doubt and find it plausible that there is a genuine SDG way of arriving at FM. Might not even be hard to come by, but I haven’t really thought about it. If you have, I’d sure welcome you adding a corresponding hint to the entry!
You’re right that I should get this sorted out. The way it looks to me, and what bugs me, is that the article confidently declares there is no relation to the sense of infinitesimal in SDG. I would prefer that the article remain silent on the issue until it gets sorted out, or else buffer the declaration with e.g. “this author (Urs Schreiber) believes…”
Maybe I’ll start with Fulton-MacPherson, not Axelrod-Singer. Fulton at least is a full-fledged algebraic geometer who can be expected to use the language of blowing-up in technically correct fashion (my possibly incorrect memory is that they deal with the complex case).
I admit that it’s possible, too, that the infinitesimals involved might be more smoothly described in the setting of nonstandard analysis (which of course can be accommodated within SDG as well).
Might the problem be from your point of view, Urs, that the classical models of SDG consist of smooth objects, whereas here we are dealing with singular configurations? There should be a theory of singular spaces as well in SDG, but I don’t know what the state of the art might be.
Sorry to go on about this, but it so happens that I’ve been making my way back into this area very recently myself, independently of what you’ve been writing up here. John recently posted something to the Café on old notes of mine (written in 1999) that touch upon general proposed constructions that include the associahedral operad, the Fulton-MacPherson operad, etc. (here), which I’m now rewriting in conversation with Samuel Vidal (and with John mostly listening in). In the first part of those notes (section A) I’m trying to describe a general formal mechanism for operadic resolutions which starting from n-simplices give the associahedra, and starting from configuration spaces give the ASFM operads, etc. It formally works for any relation you might care to give between a species and the free operad it generates.
I think there are probably lots of geometric points of view on this. From one point of view one is simply chopping off corners from polyhedra in some operadically controlled fashion (a hint I picked up from Gelfand).
Anyway, I hope to report on progress on this soon.
“this author (Urs Schreiber) believes…”
I am not expressing any belief, but am just alerting the reader of the fact that, despite common jargon, the FM-compactification is a plain topological space, not an object in a smooth topos.
As concerns my believes, I have told you in #5 and in #13 that I do find it plausible that there is an SDG construction.
I’ll bow out of this exchange now.
Urs, I can see you are getting annoyed, and I’m sorry for that, but it reads
but this is heuristics and unrelated to actual infinitesimal neighbourhoods in synthetic differential geometry
which I consider to be an overly strong statement, much stronger than you are making it out to be in #17. It does not seem at all consistent with
As concerns my beliefs, I have told you in #5 and in #13 that I do find it plausible that there is an SDG construction.
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