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It should be the case that $CRing^{op}$ is coreflective in $(CRing^{\Delta^{op}})^{op}$, the coreflection being 0-truncation of simplical rings
$CRing^{op} \stackrel{\hookrightarrow}{\underset{\tau_0}{\longleftarrow}} (CRing^{\Delta^{op}})^\op \,.$Thatâ€™s the kind of structure on sites that induces differential cohesion. Need to check some details tomorrow when I am more awake.
This must have surfaced before. For instance a derived analogue of a de Rham stack construction that does not remove nilpotent ring elements, but removes higher simplicial cells in the ring. Has this been discussed anywhere?
(Thanks to Mathieu Anel for discussion today.)
Categories of simplicial objects often have cohesive structure. Indeed, if $\mathcal{C}$ is a category of finitary algebraic structures (= locally finitely sifted-presentable) then we have
$\pi_0 \dashv \Delta \dashv \Gamma \dashv \nabla : \mathcal{C} \to [\mathbf{\Delta}^\mathrm{op}, \mathcal{C}]$with $\pi_0$ preserving finite products and $\Delta$ fully faithful. This is also true if $\mathcal{C}$ is a $\sigma$-pretopos.
Yes, but the above is about something different. Not about cohesion of simplicial objects, but about differential cohesion of simplicial sheaves over duals of simplicial rings.
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