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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 4th 2014
• (edited Nov 4th 2014)

It should be the case that $CRing^{op}$ is coreflective in $(CRing^{\Delta^{op}})^{op}$, the coreflection being 0-truncation of simplical rings

$CRing^{op} \stackrel{\hookrightarrow}{\underset{\tau_0}{\longleftarrow}} (CRing^{\Delta^{op}})^\op \,.$

That’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.)

• CommentRowNumber2.
• CommentAuthorZhen Lin
• CommentTimeNov 4th 2014

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 4th 2014
• (edited Nov 4th 2014)

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.