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nLab > Latest Changes: differentiable Lie group cohomology as intrinsic (oo,1)-topos cohomology

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMay 29th 2010
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Author: Urs
Format: MarkdownItexI 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 <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#Cohomology">this new section</a>. Towards the end I could expand still a bit more, but I am not allowed to work anymore today... :-)

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
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Author: Urs
Format: MarkdownItex> 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...

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)
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Author: Urs
Format: MarkdownItexI 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 <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#CohomologyOfLieGroups">oo-Lie groupoid -- Lie group cohomology</a>. 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 $G^{\times_k}$ a cofibrant replacement $C(\{U^k_i\}) \stackrel{\simeq}{\to} G^{\times_n}$ and then setting $$ H^n_{diffr}(G,A) := sPSh(\int^{[k]} \Delta[k] \otimes C(\{U^k_i\}) , \mathbf{B}^n A) $$ But even though each $C(\{U^k_i\})$ is cofibrant, the above totalization is, while related by a zig-zag of weak equivalences to $\mathbf{B}G$, not cofibrant. What is a cofibrant replacement for $\mathbf{B}G$ is $$ \int^{[k]} \mathbf{\Delta}[k] \otimes C(\{U^k_i\}) $$ where the fat Delta is the Bousfield-Kan resolution $\mathbf{\Delta}[k] = N([k]/\Delta[op]^{op})$ of the point in $[\Delta^{op}, sSet]_{proj}$. So the "true" intrinsic cohomology $H^n_{true}(G,A)$ of $\mathbf{B}G$ is obtained by mapping out of this bigger guy. By pullback along the [[Bousfield-Kan map]] we hence have a natural morphism $$ H^n_{naive}(G,A) \to H^n_{diffr}(G,A) \to H^n_{true}(G,A) \,. $$ First I thought I had shown that the last map is an iso due to the abelianness of $A$, but I don't see my argument anymore.

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>G</mi> <mrow><msub><mo>×</mo> <mi>k</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">G^{\times_k}</annotation></semantics></math>$ a cofibrant replacement $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msubsup><mi>U</mi> <mi>i</mi> <mi>k</mi></msubsup><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>G</mi> <mrow><msub><mo>×</mo> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">C(\{U^k_i\}) \stackrel{\simeq}{\to} G^{\times_n}</annotation></semantics></math>$ and then setting
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>H</mi> <mi>diffr</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>sPSh</mi><mo stretchy="false">(</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msubsup><mi>U</mi> <mi>i</mi> <mi>k</mi></msubsup><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^n_{diffr}(G,A) := sPSh(\int^{[k]} \Delta[k] \otimes C(\{U^k_i\}) , \mathbf{B}^n A) </annotation></semantics></math>$
But even though each $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msubsup><mi>U</mi> <mi>i</mi> <mi>k</mi></msubsup><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U^k_i\})</annotation></semantics></math>$ is cofibrant, the above totalization is, while related by a zig-zag of weak equivalences to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>$ , not cofibrant. What is a cofibrant replacement for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>$ is
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msubsup><mi>U</mi> <mi>i</mi> <mi>k</mi></msubsup><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int^{[k]} \mathbf{\Delta}[k] \otimes C(\{U^k_i\}) </annotation></semantics></math>$
where the fat Delta is the Bousfield-Kan resolution $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>op</mi><msup><mo stretchy="false">]</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Delta}[k] = N([k]/\Delta[op]^{op})</annotation></semantics></math>$ of the point in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, sSet]_{proj}</annotation></semantics></math>$ . So the “true” intrinsic cohomology $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi> <mi>true</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n_{true}(G,A)</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>$ is obtained by mapping out of this bigger guy. By pullback along the Bousfield-Kan map we hence have a natural morphism
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>H</mi> <mi>naive</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>diffr</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>true</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^n_{naive}(G,A) \to H^n_{diffr}(G,A) \to H^n_{true}(G,A) \,. </annotation></semantics></math>$
First I thought I had shown that the last map is an iso due to the abelianness of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ , but I don’t see my argument anymore.

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nLab > Latest Changes: differentiable Lie group cohomology as intrinsic (oo,1)-topos cohomology