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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 $\infty Grpd_{Glob}$ for the global equivariant homotopy theory and by its smooth version I mean
$\mathbf{H} \coloneqq Sh_\infty(SmoothMfd, \infty Grpd_{Glob}) \,.$This sits now in a commuting square of geometric morphisms, each one of which exhibits cohesion over its codomain:
$\array{ Sh_\infty(SmoothMfd, \infty Grpd_{Glob}) &\stackrel{\Gamma_{smth}}{\longrightarrow}& \infty Grpd_{Glob} \\ \downarrow^{\mathrlap{\Gamma_{glob}}} &\searrow^{\mathrlap{\Gamma}}& \downarrow \\ Sh_\infty(SmoothMfd, \infty Grpd) &\longrightarrow& \infty Grpd } \,.$This provides a more refined perspective on smooth quotient spaces: for instance for $X$ a smooth manifold equipped with the action of a group $G$, then this defines the presheaf on manifolds
$X /_{glob}G : U \mapsto \delta_{C^\infty(U,G)} (C^\infty(U,X), C^\infty(U,G)) \in \infty Grpd_{glob} \,,$where we regard $(C^\infty(U), C^\infty(U,G))$ as a topological $C^\infty(U,G)$-space (which happens to be topologically discrete in this example) and $\delta_{C^\infty(U,G)}$ regards that as a presheaf over $Glob$.
Then
$\Gamma_{glob} (X/_{glob} G)$ is the smooth orbifold coresponding to the $G$-action on $X$
$\Pi_{glob} (X/_{glob} G)$ is the diffeological quotient space of $X$ by $G$.
I think this is going to be important for the application to singular $G_2$-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.
Is there likely to be any nice way to fit the global aspect with the modalities of the process? Does $Sh_\infty(CartSp_{supersynth}, \infty Grpd_{Glob})$ provide a model for the 12 modalities?
From the above diagram we get three adjoint triples of modalities:
$\Pi \dashv \flat \dashv \sharp$and
$\Pi_{glob} \dashv \flat_{glob} \dashv \sharp_{glob}$and
$\Pi_{smth} \dashv \flat_{smth} \dashv \sharp_{smth}$And I think we have the relation
$\flat_{glob} \flat_{smth} \simeq \flat_{smth} \flat_{glob} \simeq \flat$So it’s a kind of factoring of the absolute cohesion into two subaspects.
I don’t understand the example: what does $(C^\infty(U), C^\infty(U,G))$ represent? Where is the dependence on $X$?
Oh, sorry, the worst of all typos. The $C^\infty(U)$ was meant to read $C^\infty(U,X)$. Sorry for this.
All I am saying is that a manifold with a $G$-action represents a sheaf of sets with group action, hence a sheaf of topological G-spaces.
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