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nLab > Latest Changes: Borel model structure

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeOct 26th 2020
- PermaLink
Author: Urs
Format: MarkdownItexmade some minor cosmetic edits, such as replacing \bar W G (which comes out with too short an overline) with \overline{W} G <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>
made some minor cosmetic edits, such as replacing
```
  \bar W G
```
(which comes out with too short an overline) with
```
  \overline{W} G
```
diff, v2, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeJun 4th 2021
- PermaLink
Author: Urs
Format: MarkdownItexI have made the Quillen equivalence to the slice model structure over $\overline{W}G$ a little more explicit. Also streamlined other parts of the entry a little. <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

I have made the Quillen equivalence to the slice model structure over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\overline{W}G</annotation></semantics></math>$ a little more explicit. Also streamlined other parts of the entry a little.

diff, v5, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeJun 5th 2021
- PermaLink
Author: Urs
Format: MarkdownItexI have added a remark ([here](https://ncatlab.org/nlab/show/Borel+model+structure#sSetEnrichmentOfAdjunctionToSliceOverSimpClassSpace)) making explicit that the adjunction with the slice over $\overline{W}G$ is indeed simplicial (a fact that is not quite made explicit in [Dror, Dwyer & Kan 80](https://ncatlab.org/nlab/show/Borel+model+structure#DDK80) ) <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/9">v9</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

I have added a remark (here) making explicit that the adjunction with the slice over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\overline{W}G</annotation></semantics></math>$ is indeed simplicial (a fact that is not quite made explicit in Dror, Dwyer & Kan 80 )

diff, v9, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeJun 5th 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer also to * [[Paul Goerss]], [[J. F. Jardine]], Section V.2 (Thm. 2.3) of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)) <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/10">v10</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>
added pointer also to
- Paul Goerss, J. F. Jardine, Section V.2 (Thm. 2.3) of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)
diff, v10, current
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeJun 22nd 2021
- (edited Jun 22nd 2021)
- PermaLink
Author: Urs
Format: MarkdownItexI made a note ([here](https://ncatlab.org/nlab/show/Borel+model+structure#RelationToModelStructureOnPlainSimplicialSets), still in need of polishing and proof-reading ) that for $\mathcal{G}$ any simplicial group we have a (forgetful $\dashv$ cofree)-Quillen adjunction $$ sSet \underoverset {\underset{ \;\;\; [\mathcal{G},-] \;\;\; }{\longrightarrow}} {\overset{ \;\;\; undrl \;\;\; }{\longleftarrow}} {\bot} \mathcal{G}Actions(sSet) \,. $$ The Quillen functor property is immediate from the other propositions in the entry once we know that the cofree right adjoint exists at all, and so in the note I just spell out that right adjoint. It's all tautological, of course, but I wanted to write it out because one can't quite argue pointwise as for topological $G$-spaces but needs [this formula](https://ncatlab.org/nlab/show/Borel+model+structure#eq:CofreeSimplicialActionInComponents), I think. <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/11">v11</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

I made a note (here, still in need of polishing and proof-reading ) that for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math>$ any simplicial group we have a (forgetful $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math>$ cofree)-Quillen adjunction
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sSet</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/><mo stretchy="false">[</mo><mi>𝒢</mi><mo>,</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/></mrow></munder><mover><mo>⟵</mo><mrow><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/><mi>undrl</mi><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/></mrow></mover></munderover><mi>𝒢</mi><mi>Actions</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> sSet \underoverset {\underset{ \;\;\; [\mathcal{G},-] \;\;\; }{\longrightarrow}} {\overset{ \;\;\; undrl \;\;\; }{\longleftarrow}} {\bot} \mathcal{G}Actions(sSet) \,. </annotation></semantics></math>$
The Quillen functor property is immediate from the other propositions in the entry once we know that the cofree right adjoint exists at all, and so in the note I just spell out that right adjoint. It’s all tautological, of course, but I wanted to write it out because one can’t quite argue pointwise as for topological $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ -spaces but needs this formula, I think.

diff, v11, current
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeJun 23rd 2021
- PermaLink
Author: Urs
Format: MarkdownItexhave now slightly polished-up the writeup of that proof of the cofree simplicial action Quillen adjunction ([here](https://ncatlab.org/nlab/show/Borel+model+structure#RelationToModelStructureOnPlainSimplicialSets)). Should be good now. But this ought to be textbook material. If anyone has a reference, let's add it. <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/14">v14</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

have now slightly polished-up the writeup of that proof of the cofree simplicial action Quillen adjunction (here). Should be good now. But this ought to be textbook material. If anyone has a reference, let’s add it.

diff, v14, current
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeJun 23rd 2021
- PermaLink
Author: Urs
Format: MarkdownItexI have spelled out the example ([here](https://ncatlab.org/nlab/show/Borel+model+structure#BZActionOnInertiaGroupoid)) of the canonical $\mathbf{B}\mathbb{Z}$-action on an inertia groupoid <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

I have spelled out the example (here) of the canonical $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math>$ -action on an inertia groupoid

diff, v16, current
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeJul 1st 2021
- (edited Jul 1st 2021)
- PermaLink
Author: Urs
Format: MarkdownItexadded the observation ([here](https://ncatlab.org/nlab/show/Borel+model+structure#GeneralizationToSimplicialPresheaves)) that the adjunction for simplicial groups $$ \mathcal{G} Acts(sSet) \underoverset {\underset{ \big((-) \times W \mathcal{G}\big)/\mathcal{G} }{\longrightarrow}} {\overset{ (-) \times_{\overline{W}\mathcal{G}} W \mathcal{G} }{\longleftarrow}} {\bot} sSet_{/\overline{W}\mathcal{G}} $$ generalizes to one for presheaves of simplicial groups $$ \underline{\mathcal{G}} Acts \big( sPSh(\mathcal{C}) \big) \underoverset { \underset{ \big( (-) \times W\underline{\mathcal{G}} \big) \big/ \underline{\mathcal{G}} } {\longrightarrow}} { \overset{ (-) \times_{\overline{W}\underline{\mathcal{G}}} W\underline{\mathcal{G}} }{\longleftarrow} } {\bot} sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}} $$ Maybe the notation can be improved. One needs that homomorphisms of actions of presheaves of groups are universal with respect to squares of the form $$ \array{ \underline{\mathcal{G}}Acts \big( \underline{A}, \, \underline{B} \big) &\longrightarrow& \underline{\mathcal{G}}(c_1)Acts \big( \underline{A}(c_1), \, \underline{B}(c_1) \big) \\ \big\downarrow && \big\downarrow \\ \underline{\mathcal{G}}(c_2)Acts \big( \underline{A}(c_2), \, \underline{B}(c_2) \big) &\longrightarrow& Hom \big( \underline{A}(c_1), \, \underline{B}(c_2) \big) } $$ <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/18">v18</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

added the observation (here) that the adjunction for simplicial groups
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>𝒢</mi><mi>Acts</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>W</mi><mi>𝒢</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><mi>𝒢</mi></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi></mrow></msub><mi>W</mi><mi>𝒢</mi></mrow></mover></munderover><msub><mi>sSet</mi> <mrow><mo stretchy="false">/</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{G} Acts(sSet) \underoverset {\underset{ \big((-) \times W \mathcal{G}\big)/\mathcal{G} }{\longrightarrow}} {\overset{ (-) \times_{\overline{W}\mathcal{G}} W \mathcal{G} }{\longleftarrow}} {\bot} sSet_{/\overline{W}\mathcal{G}} </annotation></semantics></math>$
generalizes to one for presheaves of simplicial groups
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub><mi>W</mi><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></mover></munderover><mi>sPSh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mover><mi>W</mi><mo>¯</mo></mover><munder><mi>𝒢</mi><mo>̲</mo></munder></mrow></msub></mrow><annotation encoding="application/x-tex"> \underline{\mathcal{G}} Acts \big( sPSh(\mathcal{C}) \big) \underoverset { \underset{ \big( (-) \times W\underline{\mathcal{G}} \big) \big/ \underline{\mathcal{G}} } {\longrightarrow}} { \overset{ (-) \times_{\overline{W}\underline{\mathcal{G}}} W\underline{\mathcal{G}} }{\longleftarrow} } {\bot} sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}} </annotation></semantics></math>$
Maybe the notation can be improved. One needs that homomorphisms of actions of presheaves of groups are universal with respect to squares of the form
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mi>𝒢</mi><mo>̲</mo></munder><mi>Acts</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo>,</mo><mspace width="0.16667em"/><munder><mi>B</mi><mo>̲</mo></munder><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>Acts</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="0.16667em"/><munder><mi>B</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd/> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mi>𝒢</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>Acts</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="0.16667em"/><munder><mi>B</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Hom</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mi>A</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="0.16667em"/><munder><mi>B</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underline{\mathcal{G}}Acts \big( \underline{A}, \, \underline{B} \big) &amp;\longrightarrow&amp; \underline{\mathcal{G}}(c_1)Acts \big( \underline{A}(c_1), \, \underline{B}(c_1) \big) \\ \big\downarrow &amp;&amp; \big\downarrow \\ \underline{\mathcal{G}}(c_2)Acts \big( \underline{A}(c_2), \, \underline{B}(c_2) \big) &amp;\longrightarrow&amp; Hom \big( \underline{A}(c_1), \, \underline{B}(c_2) \big) } </annotation></semantics></math>$
diff, v18, current
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeSep 4th 2021
- PermaLink
Author: Urs
Format: MarkdownItexmade explicit ([here](https://ncatlab.org/nlab/show/Borel+model+structure#BorelModelStructureOnTopologicalSpaces)) also the version in topological spaces (previously the entry focused on simplicial sets) <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/21">v21</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

made explicit (here) also the version in topological spaces (previously the entry focused on simplicial sets)

diff, v21, current
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeSep 4th 2021
- (edited Sep 4th 2021)
- PermaLink
Author: Urs
Format: MarkdownItexstarted ([here](https://ncatlab.org/nlab/show/Borel+model+structure#PropertiesInTopologicalSpaces)) a new subsection, recording basic properties of the projective model structure on $G Act(TopSp)$, leading up to the Borel construction as a left derived functor (For the moment almost straight from the last page of [Guillou's note](http://www.ms.uky.edu/~guillou/EquivModels.pdf), up to spelling out of some basic details that Guillou leaves implicit). <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/21">v21</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

started (here) a new subsection, recording basic properties of the projective model structure on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>Act</mi><mo stretchy="false">(</mo><mi>TopSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G Act(TopSp)</annotation></semantics></math>$ , leading up to the Borel construction as a left derived functor

(For the moment almost straight from the last page of Guillou’s note, up to spelling out of some basic details that Guillou leaves implicit).

diff, v21, current
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeSep 4th 2021
- (edited Sep 4th 2021)
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Steffen Sagave]], [[Christian Schlichtkrull]], Prop. 6.4 in: *Diagram spaces and symmetric spectra*, Advances in Mathematics, 231 (2012), 2116-2193 ([arXiv:1103.2764](https://arxiv.org/abs/1103.2764), [doi:10.1016/j.aim.2012.07.013](https://doi.org/10.1016/j.aim.2012.07.013)) <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/24">v24</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>
added pointer to:
- Steffen Sagave, Christian Schlichtkrull, Prop. 6.4 in: Diagram spaces and symmetric spectra, Advances in Mathematics, 231 (2012), 2116-2193 (arXiv:1103.2764, doi:10.1016/j.aim.2012.07.013)
diff, v24, current
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeSep 4th 2021
- (edited Sep 4th 2021)
- PermaLink
Author: Urs
Format: MarkdownItexhave spelled out the proof ([here](https://ncatlab.org/nlab/show/Borel+model+structure#SimplicialProofOfEquivalenceToSliceModelStructure)) that $$ G Act\big(sSet_{Qu}\big)_{proj} \underoverset {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}} {\overset{ (-) \times_{{}_{\overline{W}G}} W G }{\longleftarrow}} {\bot} \big(sSet_{Qu}\big)_{/\overline{W}G} $$ is a Quillen adjunction <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/29">v29</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

have spelled out the proof (here) that
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>G</mi><mi>Act</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>W</mi><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><mi>G</mi></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mrow/> <mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub></mrow></msub><mi>W</mi><mi>G</mi></mrow></mover></munderover><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mo stretchy="false">/</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> G Act\big(sSet_{Qu}\big)_{proj} \underoverset {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}} {\overset{ (-) \times_{{}_{\overline{W}G}} W G }{\longleftarrow}} {\bot} \big(sSet_{Qu}\big)_{/\overline{W}G} </annotation></semantics></math>$
is a Quillen adjunction

diff, v29, current
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeSep 17th 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded proposition and proof ([here](https://ncatlab.org/nlab/show/Borel+model+structure#BorelConstructionOfFreeActionEquivalentToPlainQuotient)) that the topological Borel construction of a free action (at least for compact Lie group $G$ acting on a $G$-CW complex) is weakly equivalent to the plain quotient <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/31">v31</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

added proposition and proof (here) that the topological Borel construction of a free action (at least for compact Lie group $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ acting on a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ -CW complex) is weakly equivalent to the plain quotient

diff, v31, current
- CommentRowNumber14.
- CommentAuthornLab edit announcer
- CommentTimeNov 20th 2021
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexG is a group, not just a space. Doron Grossman-Naples <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/34">v34</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

G is a group, not just a space.

Doron Grossman-Naples

diff, v34, current
- CommentRowNumber15.
- CommentAuthorUrs
- CommentTimeNov 21st 2021
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Author: Urs
Format: MarkdownItexThis was in the first line [here](https://ncatlab.org/nlab/show/Borel+model+structure#Idea). Thanks. I have now also fixed a grammar error further down, and added previously missing link to *[[fine model structure on topological G-spaces]]*. <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/35">v35</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

This was in the first line here. Thanks.

I have now also fixed a grammar error further down, and added previously missing link to fine model structure on topological G-spaces.

diff, v35, current
- CommentRowNumber16.
- CommentAuthorUrs
- CommentTimeApr 29th 2023
- (edited Apr 29th 2023)
- PermaLink
Author: Urs
Format: MarkdownItexadded detailed proof ([here](https://ncatlab.org/nlab/show/Borel+model+structure#MonoidalModelStructure)) of monoidal model structure on $\mathcal{G}Act(\mathbf{C})_{Borel}$, essentially by the argument of [Berger & Moerdijk (2006), Lem. 2.5.2](https://ncatlab.org/nlab/show/Borel+model+structure#BergerMoerdijk2006), but generalized to coefficients in any cofibrantly generated simplicial monoidal model category $\mathbf{C}$ (beyond just $\mathbf{C} = sSet$) and not forgetting to also check the unit axiom. <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/41">v41</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

added detailed proof (here) of monoidal model structure on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>𝒢</mi><mi>Act</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><msub><mo stretchy="false">)</mo> <mi>Borel</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}Act(\mathbf{C})_{Borel}</annotation></semantics></math>$ , essentially by the argument of Berger & Moerdijk (2006), Lem. 2.5.2, but generalized to coefficients in any cofibrantly generated simplicial monoidal model category $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>$ (beyond just $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>=</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\mathbf{C} = sSet</annotation></semantics></math>$ ) and not forgetting to also check the unit axiom.

diff, v41, current
- CommentRowNumber17.
- CommentAuthorUrs
- CommentTimeMay 17th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadjusted wording in statement and proof of the monoidalness of the model strcucture on $\mathbf{C}^{\mathbf{B} \mathcal{G}}$ ([here](https://ncatlab.org/nlab/show/Borel+model+structure#ExistenceOfMonoidalModelStructure)) for monoidal simplicial combinatorial $\mathbf{C}$ to clarify that this uses (needs?) the assumption that all objects of $\mathbf{C}$ are cofibrant. <a href="https://ncatlab.org/nlab/revision/diff/Borel+model+structure/49">diff</a>, <a href="https://ncatlab.org/nlab/revision/Borel+model+structure/49">v49</a>, <a href="https://ncatlab.org/nlab/show/Borel+model+structure">current</a>

adjusted wording in statement and proof of the monoidalness of the model strcucture on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝒢</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{C}^{\mathbf{B} \mathcal{G}}</annotation></semantics></math>$ (here) for monoidal simplicial combinatorial $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>$ to clarify that this uses (needs?) the assumption that all objects of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>$ are cofibrant.

diff, v49, current

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