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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

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2021

    I have made the Quillen equivalence to the slice model structure over W¯G\overline{W}G a little more explicit. Also streamlined other parts of the entry a little.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021

    I have added a remark (here) making explicit that the adjunction with the slice over W¯G\overline{W}G is indeed simplicial (a fact that is not quite made explicit in Dror, Dwyer & Kan 80 )

    diff, v9, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021

    added pointer also to

    diff, v10, current

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

    I made a note (here, still in need of polishing and proof-reading ) that for 𝒢\mathcal{G} any simplicial group we have a (forgetful \dashv cofree)-Quillen adjunction

    sSet[𝒢,]undrl𝒢Actions(sSet). 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 GG-spaces but needs this formula, I think.

    diff, v11, current

    • 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.

    diff, v14, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2021

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

    diff, v16, current

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

    added the observation (here) that the adjunction for simplicial groups

    𝒢Acts(sSet)(()×W𝒢)/𝒢()× W¯𝒢W𝒢sSet /W¯𝒢 \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

    𝒢̲Acts(sPSh(𝒞))(()×W𝒢̲)/𝒢̲()× W¯𝒢̲W𝒢̲sPSh(𝒞) /W¯𝒢̲ \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

    𝒢̲Acts(A̲,B̲) 𝒢̲(c 1)Acts(A̲(c 1),B̲(c 1)) 𝒢̲(c 2)Acts(A̲(c 2),B̲(c 2)) Hom(A̲(c 1),B̲(c 2)) \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) }

    diff, v18, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021

    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)

    started (here) a new subsection, recording basic properties of the projective model structure on GAct(TopSp)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).

    diff, v21, current

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

    added pointer to:

    diff, v24, current

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

    have spelled out the proof (here) that

    GAct(sSet Qu) proj(()×WG)/G()× W¯GWG(sSet Qu) /W¯G 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

    diff, v29, current

    • 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 GG acting on a GG-CW complex) is weakly equivalent to the plain quotient

    diff, v31, current

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

    Doron Grossman-Naples

    diff, v34, current

    • 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.

    diff, v35, current

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