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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2010

    I moved the proof of the claim that the Segal-Brylinski “differetiable Lie group cohomology” is that computed in the (oo,1)-topos of oo-Lie groupoids from the entry group cohomology to the entry Lie infinity-groupoid and expanded the details of the proof considerably.

    See this new section.

    Towards the end I could expand still a bit more, but I am not allowed to work anymore today… :-)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2010

    Towards the end I could expand still a bit more,

    Hm, in fact I think I have a gap towards the end. So far it just shows an inclusion of Segal-Brylinski cohomology into the (oo,1)-topos cohomology, not an isomorphisms.

    Hm, darn, I need to think more about it…

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2010
    • (edited May 31st 2010)

    I don’t see the equivalence anymore that I claimed originally, it now seems to me that the intrinsic (oo,1)-topos Lie group cohomology is even finer than the Segal-Brylinski differential cohomology (which refines the “naive” Lie group cohomology).

    I adjusted the entries accordingly, the details are at oo-Lie groupoid – Lie group cohomology.

    But maybe to highlight the sensitive point within the discussion there:

    Brylinski’s differentiable Lie group cohomology is obtained (in paraphrase) by choosing funcorially for each G × kG^{\times_k} a cofibrant replacement C({U i k})G × nC(\{U^k_i\}) \stackrel{\simeq}{\to} G^{\times_n} and then setting

    H diffr n(G,A):=sPSh( [k]Δ[k]C({U i k}),B nA) H^n_{diffr}(G,A) := sPSh(\int^{[k]} \Delta[k] \otimes C(\{U^k_i\}) , \mathbf{B}^n A)

    But even though each C({U i k})C(\{U^k_i\}) is cofibrant, the above totalization is, while related by a zig-zag of weak equivalences to BG\mathbf{B}G, not cofibrant. What is a cofibrant replacement for BG\mathbf{B}G is

    [k]Δ[k]C({U i k}) \int^{[k]} \mathbf{\Delta}[k] \otimes C(\{U^k_i\})

    where the fat Delta is the Bousfield-Kan resolution Δ[k]=N([k]/Δ[op] op)\mathbf{\Delta}[k] = N([k]/\Delta[op]^{op}) of the point in [Δ op,sSet] proj[\Delta^{op}, sSet]_{proj}. So the “true” intrinsic cohomology H true n(G,A)H^n_{true}(G,A) of BG\mathbf{B}G is obtained by mapping out of this bigger guy. By pullback along the Bousfield-Kan map we hence have a natural morphism

    H naive n(G,A)H diffr n(G,A)H true n(G,A). H^n_{naive}(G,A) \to H^n_{diffr}(G,A) \to H^n_{true}(G,A) \,.

    First I thought I had shown that the last map is an iso due to the abelianness of AA, but I don’t see my argument anymore.

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