Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration-theory internal-categories k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    starting some minimum

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018
    • (edited Nov 3rd 2018)

    added more references:


    [ operad structured due to Getzler-Jones… ]

    and alternatively in

    which was corrected in

    • Giovanni Gaiffi, Models for real subspace arrangements and stratified manifolds, Int. Math. Res. Not. 12:627-656, 2003 (pdf)

    and developed in detail in

    • Dev Sinha, Manifold theoretic compactifications of configuration spaces, Selecta Math. (N.S.) 10(3):391-428, 2004 (arXiv:math/0306385)

    which also shows the equivalence to Fulton-MacPherson 94.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    spelled out the definition of the Fulton-MacPherson compactification as a space (not yet the operad-structure) and added an Idea section with an informal outline

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 3rd 2018

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    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?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    added a section (here) indicating the now common pictorial notation of the boundary points by “funnels” (I’d suggest better: magnifying glasses!)

    diff, v4, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    added statement (here) of the presentation of the deRham cohomology of FM n(d)FM_n(d) due to Cohen 73, in terms of the volume forms of the boundary spheres (i.e. of the “exceptional divisors”, I suppose)

    diff, v5, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    added statement (here) of the quasi-iso from the graph complex to the de Rham cohomology of the FM-compactification from Lambrechst-Volic 12, theorem 8.1

    this wants to be exapnded further. But I’ll take a break now

    diff, v5, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    added statement (here) of the quasi-isomorphism between the graph complex and the algebraic de Rham complex of the FM-compactification, the main result of Lambrechts-Volic 14

    diff, v7, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    added pointer to

    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.

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 4th 2018
    • (edited Nov 5th 2018)

    If the desired space is being denoted M[V]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.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2018

    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!

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 5th 2018

    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).

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 5th 2018

    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.

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 5th 2018

    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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2018

    “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.

    • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 5th 2018
    • (edited Nov 5th 2018)

    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.

    • CommentRowNumber19.
    • CommentAuthorTodd_Trimble
    • CommentTime3 days ago

    Adjusted some phrasing in accordance with prior discussion; typos.

    diff, v16, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)