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 bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 infinity integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monads monoid monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages 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 science 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 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
    • CommentTimeJan 20th 2010
    This comment is invalid XML; displaying source. <p>I added three more references to <a href="http://ncatlab.org/nlab/show/Bredon+cohomology">Bredon cohomology</a>.</p> <p>two of them, by H. Honkasolo, discuss a sheaf-cohomology version of Bredon cohomology, realized as the cohomology of a topos built from the <a href="http://ncatlab.org/nlab/show/orbit+category">orbit category</a>.</p> <p>It's too late for me today now to follow this up in detail, but I thought this might be of interest in the light of our discussion at <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=650&page=1">G-equivariant stable homotopy theory</a>.</p>
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 20th 2010
    • (edited Jan 20th 2010)

    An advice: It would be good when citing references to really cite them, not only put a blind link to the paper. Especially for offline usage and usage via printouts. I am often doing repairing these blind "citations" and done it this time for the Honkasolo references, but this procedure doubles both the effort, the bandwidth (second person needs also to download the paper to read the reference) and multiplies nlab versions. The link to second Honkasolo reference even frose/killed my browser so I did not succeed to cite it in the first attempt.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2010

    okay.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 20th 2010
    • (edited Jan 20th 2010)

    I added the redirect complex cobordism to complex cobordism cohomology theory as one usually just simply says complex cobordism, just like one says complex K-theory for something what pedantically can be called complex K-theory cohomology theory. It seems to me that Honkasolo's references have some overlap with Moerdijk-Svensson approach -- the Grothendieck construction here also plays a role.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2014
    • (edited Apr 16th 2014)

    I have added an actual Definition-section to Bredon cohomology.

    (Of course this overlaps with what ages ago Mike added in the subsection on Bredon cohomology at equivariant cohomology.)

    For one I mentioned the chain complex model that Bredon actually wrote down (as opposed to all the high-powered generalizations that now carry his name).

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 16th 2016

    I’m confused by the Definition at Bredon cohomology. What is H\mathbf{H} and what is BG\mathbf{B}G? We appear to be defining “H /BG\mathbf{H}_{/\mathbf{B}G}” to be the (,1)(\infty,1)-topos of GG-spaces, but that notation looks like it is a slice category of something, whereas neither of the definitions given (L fpweGTopL_{fpwe}G Top and PSh (Orb G)PSh_\infty(Orb_G)) are slices.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 17th 2016

    Yes, that notation is bad, I should change it. How about H orb G\mathbf{H}^{orb_G}?

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeAug 17th 2016

    That’s better, although it looks like a functor category, and if H\mathbf{H} means Gprd\infty Gprd then we ought to say contravariant functors, right?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2016

    Right, sure. I have changed the notation in the entry now.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 18th 2016

    Is there something from Mike’s comments here worth including? It’s in the context of a discussion of homology in enriched categories, which seems to work generally for semicartesian enrichment:

    Note that in the case when VV is cartesian monoidal and also extensive with an indecomposable terminal object, we can identify VV-enriched categories with certain VV-internal categories. In that case, this construction amounts to taking the internal nerve Cat(V)V Δ opCat(V) \to V^{\Delta^{op}} and then applying the Yoneda embedding VSet V opV \to Set^{V^{op}} levelwise to get a presheaf of simplicial sets on VV.

    In particular, therefore, if VV has a small dense subcategory V 0V_0, we can use the restricted Yoneda embedding VSet V 0 opV \to Set^{V_0^{op}} instead without losing information. This is a fancier way of saying why we see nothing new when V=SetV=Set, because the single object 11 is dense in SetSet, and its restricted Yoneda embedding is the identity.

    Another interesting example of this sort is V=GSetV=G Set for a group GG, with V 0V_0 the category of orbits G/HG/H. Thus, a category enriched over GG-sets has a homology indexed by orbits, which I think is just the usual equivariant (Bredon) homology of its GG-equivariant nerve. (Curiously, I don’t think I recall ever seeing the “morphism-graded” version in equivariant homotopy theory. Maybe it’s present implicitly somewhere?)

    Is this ’the’ reason why we see Orb GOrb_G?

    By the way, presumably the ’org’ is ’orb’, and there should be consistency as to ’orb’ or ’Orb’.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2016

    presumably the ’org’ is ’orb’

    Oh, sorry. Fixed now.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeAug 18th 2016

    I don’t think that’s “the” reason we see Orb GOrb_G, because when V=GSetV = G Set the single object G/eG/e is already dense (since it’s the unique representable when GSetG Set is regarded as a presheaf category. I’ve never really seen a convincing argument for any “the” reason why we see Orb GOrb_G.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2016
    • (edited Aug 20th 2016)

    I’ve never really seen a convincing argument for any “the” reason why we see Orb GOrb_G.

    Isn’t “the” reason given by the GG-Whitehead theorem + Elmendorf’s theorem ?

    1. There is an evident notion of homotopy equivalence between GG-CW-complexes.

    2. by the G-Whitehead theorem this is equivalent to maps that are weak homotopy equivalences on all HH-fixed point spaces for all closed subgroups HH;

    3. by Elmendorf’s theorem this is equivalent to weak equivalences in the functor category over Orb GOrb_G.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeAug 20th 2016

    Sort of, but you still have to motivate your particular definition of GG-CW-complex, don’t you?

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 3rd 2016

    Is there any sort of generalisation of Bredon cohomology from action groupoids to more general (nice) groupoids, say locally quotient groupoids? That is, does the abstract infinity-topos viewpoint allow something to be constructed?

    Carla Farsi told me that she and some others tried to develop this for proper Lie groupoids more or less by hand (or so I gather) without much success.

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeDec 5th 2016

    Orbifold cohomology is related to Bredon cohomology, see works of Ruan et al.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2020
    • (edited Jun 14th 2020)

    added these pointers to discussion of equivalence of Bredon cohomology of topological G-spaces XX to abelian sheaf cohomology of the topological quotient space X/GX/G with coefficients a “locally constant sheaf except for dependence on isotropy groups”:

    diff, v30, current

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 14th 2020

    “locally constant sheaf except for dependence on isotropy groups”

    so a constructible sheaf?

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2020

    Ah, right, thanks. Haven’t thought about this.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 14th 2020

    Some remarks:

    • Segal has a similar sheaf in his equivariant K-theory paper (prop 5.3), but it’s the sheafification of a presheaf defined using K-theory
    • The quotient of a manifold by a proper Lie group action (so with compact stabilisers) is nicely stratified by manifolds, and more generally so is the orbit space of a proper Lie groupoid
    • the constructibility will be with respect to this stratification.
    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2020

    If you want to go down that road now, allow me remark that, to my mind, the take-away message from traditional orbifold cohomology – references is, in contrast, that none of this business with funny coefficient sheaves on funny quotients is necessary if we’d just work with Bredon cohomology from the get-go. That’s what Honkasalo’s result says. And if we don’t (mis?)use sheaves on funny auxiliary space to encode genuine equivariance, we still have them available to do their real work on the geometrically meaningful spaces: namely to provide enhancement to differential equivariant cohomology.

    Of course I don’t know what piques your interest here. If you are just intrigued that Honkasalo’s and maybe Segal’s and other old results might be prettfied by being recast in the language of constructible sheaves then that might be worthwhile. On the other hand, you went to the thread on Bredon cohomology with this comment, so I am not sure what you are after.(?)

    • CommentRowNumber22.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 15th 2020
    • (edited Jun 15th 2020)

    Nothing in particular. I just happened to see the phrase I quoted in #18 in this thread (I haven’t looked at the other recent, related, threads yet), and wanted to point you to what it seemed to be referring. If you aren’t going down that path then it’s just going to be a side comment in the story you are hoping to tell, that’s all. But thanks for clarifying

    the take-away message from traditional orbifold cohomology – references is, in contrast, that none of this business with funny coefficient sheaves on funny quotients is necessary if we’d just work with Bredon cohomology from the get-go. That’s what Honkasalo’s result says.

    I had seen the ’funny coefficient sheaves’ in something I was thinking about on and off, in a way that seems rather useful, but maybe I should be thinking about Bredon cohomology (or rather genuinely global equivariant cohomology) instead.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2020

    I see. If that subsection on traditional orbifold cohomology does not leave the reader with a deep feeling of dissatisfaction, then I may need to rewrite it ;-)

    • CommentRowNumber24.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 15th 2020

    So here’s the question: what about in the non-global quotient case, as in the second half of my #15 above?

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2020

    Yeah, that’s what I had sketched out at orbifold cohomology. Polished writeup should be available soon.

    • CommentRowNumber26.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 15th 2020
    • (edited Jun 15th 2020)

    OK, thanks! My interest is more in the/a K-theory version, so that might be something that we can discuss later/elsewhere.

    On a separate note, have you recorded a reference to Schwede’s Global homotopy theory course on YouTube anywhere?

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2020

    No, I didn’t know that there are such lectures on YouTube.

    • CommentRowNumber28.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 15th 2020

    It’s working from his monograph Global homotopy theory. I will add the lectures there as a reference.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2020

    My interest is more in the/a K-theory version

    I should add pointers to traditional orbifold K-theory to the entry traditional orbifold cohomology – references.

    Adem-Ruan in “Twisted Orbifold K-Theory” arXiv:math/0107168 don’t really define orbifold K-theory. What they really do is (Def. 3.4) say that on global quotient orbifolds of some XX by some GG it should equal the genuine (Bredon) GG-equivariant K-theory of XX, and then they run with that as a definition in that case.

    Then there is the constructions by Bunke-Schick and by Freed et al. Should add commented pointers to these…

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2020
    • (edited Oct 1st 2020)

    I have added full publication data to

    This article stands out in that it admits that Bredon cohomology is equivalently homs into Eilenberg-MacLane GG-spaces.

    Is there any other reference that would expand on this? Greenlees-May just state it in passing.

    diff, v31, current

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2020

    ah, never mind, I see that it’s Theorem 2.11 in Bredon 67

    diff, v31, current

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2020

    added pointer to

    diff, v35, 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)