Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft 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
    • CommentTimeDec 4th 2010

    created stub for differential fivebrane structure

    sounds easy, but due to lots of software trouble that took me a good bit of the afternoon! :-(

  1. Hi Urs,

    I guess in the last commutative diagram at the very end of differential fivebrane structures the upper left corner should be Fivebrane and not String, right?

    Also, you know (since we have already discussed this about twisted cohomology) that I don’t like the rightmost “picking a representative” map very much. I’ll try thinking about a reformulation which avoids this.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2010

    right?

    yes, thanks, fixed.

    Also, you know (since we have already discussed this about twisted cohomology) that I don’t like the rightmost “picking a representative” map very much. I’ll try thinking about a reformulation which avoids this.

    If you can see another way to do this, that would be interesting. To me it seems the question would be to see to which extent the choice of points in connected components can be made -functorially.

  2. it seems to me we are defining the 6-groupoid of twisted differential fivebrane sructures as the set of all homotopy fibers of Hconn(X,BString)Hdiff(X,B7U(1)). This means that for each point in the target we consider the homotopy fiber of this morphism instead of the “actual” fiber. but the homotopy fiber is the “actual fiber” of a fibrant replacement, so the collection of all homotopy fibers is the collection of all fibers of a fibrant replacement. and this leads me to thinking that Fivebranediff,tw(X) could be more intrinsically defined as a fibrant replacement of 16ˆp2:Hconn(X,BString)Hdiff(X,B7U(1)).

  3. I’ve looked at the twisted cohomology entry. In the idea section it is written

    Given a morphism f:ˆBB in an (∞,1)-topos H, regarded as exhibiting a characteristic class on ˆB with values in B, the f-twisted cohomology at stage X is the fiber of f over a given element of B.This is such that if f:ˆB* is the terminal morphism, f-twisted cohomology is precisely the ordinary cohomology in H with coefficients in ˆB.

    I’d like to replace that with

    Given a morphism f:ˆBB in an (∞,1)-topos H, regarded as exhibiting a characteristic class on ˆB with values in B, the coefficient object of f-twisted cohomology is a (any) fibrant replacement of f:ˆBB. This is such that if ˆB is fibrant and f:ˆB* is the terminal morphism, f-twisted cohomology is precisely the ordinary cohomology in H with coefficients in ˆB.

    what do you think?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2010
    • (edited Dec 6th 2010)

    Hi Domenico,

    I am not sure in which sense you think of anything using the term “fibrant replacement” as being more intrinsic. The notion of fibrant replacement is kind of by definition not intrinsic to -category theory, but a hallmark of working in a model. And it depends on the model.

    Particularly from the point of view of the -category the fibrant replacemnt of any morphism is equivalent to that morphism, hence it seems to me there is no gain in defining twisted cohomology as the fibrant replacement of something.

    What does have intrinsic meaning is the notion of homotopy fiber, fibration sequence and so on. For AB a morphism in an -category we can ask if it is the homotopy fiber of some morphism BC. And in applications this condition comes close to requiring AB to be a “fibration” in the intuitive sense. For instance if AB is a principal -bundle, then it is also a homotopy fiber. All up to equivalence, of course.

    I know that the idea that twisted cohomology and hence differential cohomology is defined in terms of homotopy fibers that are written down by choosing basepoints in each connected component tends to make people feel uneasy. But I am not sure that there is anything to be worried about. For homotopy fibers over any one base point I never saw any one feeling uneasy about the fact that this base point is only one of many in its connected component. Here for twisted cohomology we are simply taking homotopy fibers over all possible connected components, so we choose a point in each connected component. But that choice of point is irrelevant: the corresponding homotopy fibers do not depend on it, up to equivalence.

    Maybe I am wrong, but from this point of view the definition appears entirely general: we have a morphism AB and want to record all its homotopy fibers. Up to equivalence, there is one per connected component of B. So we choose one basepoint in each connected component and compute the homotopy fiber over that. Nothing depends on the choices made.

    Moreover, nobody ever show himself or herself worried about this if we do this over just a single connected component and regard B as a pointed object. All that is happening here is that we take the liberty of regarding B as a pointed object in all possible ways. That should not be any more worriesome. In fact, that means that we are making fewer choices than if we equipped B with a fixed structure of a pointed object.

    For all this it seems to me to be a very natural thing to consider homotopy pullbacks of sections τ0AA to the projections π0:Aτ0A. But of course this feeling of mine is not a proof that there is not a better definition avaialble for the construction considered at twisted cohomology.

    • CommentRowNumber7.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 6th 2010
    • (edited Dec 6th 2010)

    I see I was confused about this.. :(

    yet, I still think we should avoid this choosing of points and think in more general terms: what I’m trying to say (but still in a very imprecise way) is that the twisted cohomology of f:ˆBB should be the “homotopy fibre functor” defined by this morphism, rather than its value on a particular morphism (e.g., on a section τ0BB).

    edit: and indeed we already have this point of view at twisted cohomology: it is where it is described in terms of H/B!

    edit: so in the end I’d say just this: twisted cohomology of f:ˆBB is the cohomology with coefficient object f in the oo-over category H/B

  4. also, the H/B point of view seems to simplify the discussion at “Examples. Sections as twisted functions…” in twisted cohomology. Namely the relevany morphism there is the representation ρ:BGVect, and one is considering twisted cohomology H/Vect(,ρ); in particular the twisted cohomology of constk is the space of sections as discussed there.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2010

    Hi Domenico,

    you are right, the over-category definition of twisted cohomology that we once noted there is nice. (I went now to the entry and polished it slightly).

    In terms of that, the statement that we were talking about here amounts to saying that there are many possible lifts of XH to an object in HB and that H/B(XB,f) depends only on the connected component of these lifts.

    Does this give a way to circumvent the issue that you were trying to avoid?

  5. Hi Urs,

    I see the difference between our points of view boild down to the following: I’m insisting that twisted cohomology is the cohomology of a cocycle c:XB, i.e., is H/B(c,f), whereas you are interested in twisted cohomology of the object X as the collection of all cohomologies H/B(c,f) with cH(X,B). And from your point of view no doubt that seeing that H/B(c,f) only depends on [c] in H(X,B) is a valuable point, reducing the quite badly defined “collection of all cohomologies” to the pullback along a section of H(X,B)H(X,B).

    Yet, if we stress the H/B(,f) point of view, this “collection of all cohomologies” could be seen as the graph of a functor defined on H(X,B). This comes equipped with a natural projection to H(X,B) and seems to me a more intrinsic construction than picking representatives.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2010

    Domenico,

    you are right, you have a very good point. What you emphasize indeed looks like a point of view that is better in several respects. I need to think about this a bit.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2010

    One important aspect about the definition of twisted cohomology in the form I used to use it is that it gives the correct long fiber sequence. (The one that I call the differential fiber sequence here in the context of differential cohomology). I should maybe think about how to express that in terms of H/B(,f).

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2010

    Domenico,

    another aspect that I believe you had been very right about and which I still need to sort out for myself is the intrinsic definition of BGconn. It fits into

    H(X,BGconn)iH(X,BniU(1)diff)H(X,BG)iH(X,BniU(1))

    and you suggested that it should be characterized by being the universal object with that property. That certainly sounds plausible. My problem is that I am having trouble computing the -pullback of this diagram because our standard model of the bottom morphism by BGdiffBnU(1)diff is not a fibraiton. So the -pullback is not manifestly just the ordinary pullback. But it ought to be for BGconn to come out right.

    At differential string structure is described how to resolve the bottom morphism to a fibration in the case that we have just a single curvature class, by using the model from SSSIII. That works fine for constructing differential string structures. But I am not sure yet if or how this can be adapted to get the universal property of BGconn.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2011

    I have reworked the idea-section and definition section at differential fivebrane structure slightly, such as to make full use of the material that is now at smooth infinity-groupoid. In fact, I have verbatim copied over the stuff from differential string structure and just made adjustments in the degrees and prefactors.

    Also, the writeup of statement and proof of the factorization of the differential Pontryagin class into a weak equivalence followed by a fibration that I now have at “differential string structure” goes through exactly verbatim now for the fivebrane structures. But I haven’t copied that over yet.