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nLab > Latest Changes: Borel model structure

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 26th 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 Bℤ\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

  14. 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

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023
    • (edited Apr 29th 2023)

    added detailed proof (here) of monoidal model structure on 𝒢Act(C) Borel\mathcal{G}Act(\mathbf{C})_{Borel}, essentially by the argument of Berger & Moerdijk (2006), Lem. 2.5.2, but generalized to coefficients in any cofibrantly generated simplicial monoidal model category C\mathbf{C} (beyond just C=sSet\mathbf{C} = sSet) and not forgetting to also check the unit axiom.

    diff, v41, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2023

    adjusted wording in statement and proof of the monoidalness of the model strcucture on C B𝒢\mathbf{C}^{\mathbf{B} \mathcal{G}} (here) for monoidal simplicial combinatorial C\mathbf{C} to clarify that this uses (needs?) the assumption that all objects of C\mathbf{C} are cofibrant.

    diff, v49, current

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