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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 25th 2020

made some minor cosmetic edits, such as replacing

  \bar W G


(which comes out with too short an overline) with

  \overline{W} G

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 4th 2021

I 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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 5th 2021

I have added a remark (here) 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 )

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 5th 2021

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 22nd 2021
• (edited Jun 22nd 2021)

I made a note (here, 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, I think.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 23rd 2021

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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJun 23rd 2021

I have spelled out the example (here) of the canonical $\mathbf{B}\mathbb{Z}$-action on an inertia groupoid

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 1st 2021
• (edited Jul 1st 2021)

$\mathcal{G} Acts(sSet) \underoverset {\underset{ \big((-) \times W \mathcal{G}\big)/\mathcal{G} }{\longrightarrow}} {\overset{ (-) \times_{\overline{W}\mathcal{G}} W \mathcal{G} }{\longleftarrow}} {\bot} sSet_{/\overline{W}\mathcal{G}}$

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

$\array{ \underline{\mathcal{G}}Acts \big( \underline{A}, \, \underline{B} \big) &\longrightarrow& \underline{\mathcal{G}}(c_1)Acts \big( \underline{A}(c_1), \, \underline{B}(c_1) \big) \\ \big\downarrow && \big\downarrow \\ \underline{\mathcal{G}}(c_2)Acts \big( \underline{A}(c_2), \, \underline{B}(c_2) \big) &\longrightarrow& Hom \big( \underline{A}(c_1), \, \underline{B}(c_2) \big) }$
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 4th 2021

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

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeSep 4th 2021
• (edited Sep 4th 2021)

started (here) 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, up to spelling out of some basic details that Guillou leaves implicit).

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 4th 2021
• (edited Sep 4th 2021)

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 4th 2021
• (edited Sep 4th 2021)

have spelled out the proof (here) that

$G Act\big(sSet_{Qu}\big)_{proj} \underoverset {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}} {\overset{ (-) \times_{{}_{\overline{W}G}} W G }{\longleftarrow}} {\bot} \big(sSet_{Qu}\big)_{/\overline{W}G}$

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeSep 17th 2021

added proposition and proof (here) 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

1. G is a group, not just a space.

Doron Grossman-Naples

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeNov 21st 2021

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.