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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 13th 2011
   • (edited Feb 13th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYou may or may not recall the observation, recorded at [[Lie group cohomology]], that there is a natural map from the Segal-Blanc-Brylinski refinement of Lie group cohomology to the intrinsic cohomology of Lie groups when regarded as [[smooth infinity-groupoid]]s. For a while i did not know how to see whether this natural map is an equivalence, as one would hope it is. The subtlety is that the Cech-formula that Brylinski gives for refined Lie group cohomology corresponds to making a degreewise cofibrant replacement of $\mathbf{B}G$ in $Smooth \infty Grpd$ and then taking the diagonal, and there is no reason that this diagonal is itself still cofibrant (and I don't think it is). So while Segal-Brylinski Lie group cohomology is finer and less naive than naive Lie group cohomology, it wasn't clear (to me) that it is fine enough and reproduces the "correct" cohomology. So one had to argue that for certain coefficients the degreewise cofibrant resolution in $[CartSp^{op}, sSet]_{proj,loc}$ is already sufficient for computing the [[derived hom space]]. It was only yesterday that I realized that this is a corollary of the general result at [[function algebras on infinity-stacks]] once we embed [[smooth infinity-groupoid]] into [[synthetic differential infinity-groupoid]]s. So I believe I have a proof now. I have written it out in [[synthetic differential infinity-groupoid]] in the section <a href="http://nlab.mathforge.org/nlab/show/synthetic+differential+infinity-groupoid#StrucCohomology">Cohomology and principal $\infty$-bundles</a>.

   You may or may not recall the observation, recorded at Lie group cohomology, that there is a natural map from the Segal-Blanc-Brylinski refinement of Lie group cohomology to the intrinsic cohomology of Lie groups when regarded as smooth infinity-groupoids.

   For a while i did not know how to see whether this natural map is an equivalence, as one would hope it is. The subtlety is that the Cech-formula that Brylinski gives for refined Lie group cohomology corresponds to making a degreewise cofibrant replacement of $\mathbf{B}G$ in $Smooth \infty Grpd$ and then taking the diagonal, and there is no reason that this diagonal is itself still cofibrant (and I don’t think it is). So while Segal-Brylinski Lie group cohomology is finer and less naive than naive Lie group cohomology, it wasn’t clear (to me) that it is fine enough and reproduces the “correct” cohomology.

   So one had to argue that for certain coefficients the degreewise cofibrant resolution in $[CartSp^{op}, sSet]_{proj,loc}$ is already sufficient for computing the derived hom space. It was only yesterday that I realized that this is a corollary of the general result at function algebras on infinity-stacks once we embed smooth infinity-groupoid into synthetic differential infinity-groupoids.

   So I believe I have a proof now. I have written it out in synthetic differential infinity-groupoid in the section Cohomology and principal $\infty$-bundles.