Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural nforum nlab nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)