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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2015
    • (edited Nov 24th 2015)

    It’s time that I think a bit more about the combination of smooth cohesion with Charles Rezk’s global equivariant cohesion. Here are some simple thoughts, nothing deep, just to warm up.

    I’ll write Grpd Glob\infty Grpd_{Glob} for the global equivariant homotopy theory and by its smooth version I mean

    HSh (SmoothMfd,Grpd Glob). \mathbf{H} \coloneqq Sh_\infty(SmoothMfd, \infty Grpd_{Glob}) \,.

    This sits now in a commuting square of geometric morphisms, each one of which exhibits cohesion over its codomain:

    Sh (SmoothMfd,Grpd Glob) Γ smth Grpd Glob Γ glob Γ Sh (SmoothMfd,Grpd) Grpd. \array{ Sh_\infty(SmoothMfd, \infty Grpd_{Glob}) &\stackrel{\Gamma_{smth}}{\longrightarrow}& \infty Grpd_{Glob} \\ \downarrow^{\mathrlap{\Gamma_{glob}}} &\searrow^{\mathrlap{\Gamma}}& \downarrow \\ Sh_\infty(SmoothMfd, \infty Grpd) &\longrightarrow& \infty Grpd } \,.

    This provides a more refined perspective on smooth quotient spaces: for instance for XX a smooth manifold equipped with the action of a group GG, then this defines the presheaf on manifolds

    X/ globG:Uδ C (U,G)(C (U,X),C (U,G))Grpd glob, X /_{glob}G : U \mapsto \delta_{C^\infty(U,G)} (C^\infty(U,X), C^\infty(U,G)) \in \infty Grpd_{glob} \,,

    where we regard (C (U),C (U,G))(C^\infty(U), C^\infty(U,G)) as a topological C (U,G)C^\infty(U,G)-space (which happens to be topologically discrete in this example) and δ C (U,G)\delta_{C^\infty(U,G)} regards that as a presheaf over GlobGlob.


    • Γ glob(X/ globG)\Gamma_{glob} (X/_{glob} G) is the smooth orbifold coresponding to the GG-action on XX

    • Π glob(X/ globG)\Pi_{glob} (X/_{glob} G) is the diffeological quotient space of XX by GG.

    I think this is going to be important for the application to singular G 2G_2-compactifications of 11d supergravity. There one needs smooth spaces with conical singularities of ADE type, but the actual physical manifold is not supposed to be the ADE orbifold, but really the naive quotient with that singularity.

    In fact what one really wants is that one considers the singular quotient in complex analytic cohesion and then blows up the singularity, replacing the singular point by a system of spheres that touch each other such as to form the corresponding ADE Dynkin diagram. I am wondering if there is any way to capture this abstractly.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 24th 2015

    Is there likely to be any nice way to fit the global aspect with the modalities of the process? Does Sh (CartSp supersynth,Grpd Glob)Sh_\infty(CartSp_{supersynth}, \infty Grpd_{Glob}) provide a model for the 12 modalities?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2015
    • (edited Nov 24th 2015)

    From the above diagram we get three adjoint triples of modalities:

    Π \Pi \dashv \flat \dashv \sharp


    Π glob glob glob \Pi_{glob} \dashv \flat_{glob} \dashv \sharp_{glob}


    Π smth smth smth \Pi_{smth} \dashv \flat_{smth} \dashv \sharp_{smth}

    And I think we have the relation

    glob smth smth glob \flat_{glob} \flat_{smth} \simeq \flat_{smth} \flat_{glob} \simeq \flat

    So it’s a kind of factoring of the absolute cohesion into two subaspects.

    • CommentRowNumber4.
    • CommentAuthorCharles Rezk
    • CommentTimeNov 24th 2015
    • (edited Nov 24th 2015)

    I don’t understand the example: what does (C (U),C (U,G))(C^\infty(U), C^\infty(U,G)) represent? Where is the dependence on XX?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2015

    Oh, sorry, the worst of all typos. The C (U)C^\infty(U) was meant to read C (U,X)C^\infty(U,X). Sorry for this.

    All I am saying is that a manifold with a GG-action represents a sheaf of sets with group action, hence a sheaf of topological G-spaces.