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Am giving this its own page, for a coherent discussion, and to supercede the scattered remarks in various entries.
Meaning to amplify that the cohesion is a direct consequence of the fact that the $G$-orbit category is reflective in the $\prec\!\! G$-slice of the $(2,1)$-category of delooping groupoids.
Not done yet, though, but need to save.
Does anything interesting emerge from the differential cohomology hexagon in these cases?
I am not sure what to make of the differential cohomology hexagon for the “singular” version of cohesion. This is related to the fact that here the conceptual meaning of the modalities is rather different from that in “smooth” cohesion. For instance the singular-cohesive analog of the shape modalty is (speaking in the slice over $\prec \!\! 1$) the operation that sends an orbispace to its naive quotient space (while the analog of the sharp modality sends a smooth homotopy quotient to its orbi-singularization). This means that in the corresponding hexagon, the vertices now have entirely different “meaning” from what they used to have relative to smooth cohesion. Somebody needs to figure out what this meaning of the singular-cohesive hexagon (which certainly exists) is.
On the other hand, for $\mathbf{H}$ a cohesive $\infty$-topos, its globally equivariant version $Glo \mathbf{H}$ is still “smooth cohesive” over $Glo Grpd_\infty$, and the hexagon with respect to that dimension of cohesion expresses “naive” equivariant differential cohomology (this is not so naïve really, it’s Bredon-type equivariance after all and e.g. equivariant K-theory is in here, but, for better or worse, “naïve” has become a technical term here, much like “perverse” or “sober” elsewhere).
Made the main proof (here) more explicit by including pointer to the new explicit proof at slice of presheaves is presheaves on slice.
added a Lemma (here) making explicit that the left adjoint morphism of $(2,1)$-sites preserves finite products after extension to free coproduct completions.
This is under the assumption of discrete equivariance groups, using that here the $G$-orbit category is just the connected objects inside all $G$-sets, so that the extension of the left adjoint to free coproduct completions is just the 0-truncation functor $\tau_0$ in the slice $Grpd_{/B G}$, and hence preserves finite products since all $n$-truncation operations on higher toposes do.
With this it follows “formally” that the leftmost adjoint in the adjoint quadruple preserves finite products. Charles Rezk in his note works harder (in section 7.3, around p. 23) to deduce this for $G$ a compact Lie group.
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