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I have started making notes at differential cohesion on the axiomatic formulation of
So far just the bare basics. To be expanded…
The basic observation (easy in itself, but fundamental for the concept formation) is that for any differential cohesive homotopy type $X$, the inclusion of the formally étale maps into $X$ into the full slice over $X$ is not only reflective but also co-reflective (since the formally étale maps are the Pi_inf-closed morphisms with the infinitesimal path groupoid functor / de Rham space functor $\Pi_{inf}$ being a left adjoint).
This means that for $G$ any differential cohesive $\infty$-group with the corresponding de Rham coefficient object $\flat_{dR}\mathbf{B}G$ (the universal moduli for flat $\mathfrak{g}$-valued differential forms), the sheaf of flat $\mathfrak{g}$-valued forms over any $X$ is given by the sections of the coreflection of the product projection $X \times \flat_{dR}\mathbf{B}G \to X$ into the formally étale morphisms into $X$.
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