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

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

    CRing opτ 0(CRing Δ op) op. 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 𝒞\mathcal{C} is a category of finitary algebraic structures (= locally finitely sifted-presentable) then we have

    π 0ΔΓ:𝒞[Δ op,𝒞]\pi_0 \dashv \Delta \dashv \Gamma \dashv \nabla : \mathcal{C} \to [\mathbf{\Delta}^\mathrm{op}, \mathcal{C}]

    with π 0\pi_0 preserving finite products and Δ\Delta fully faithful. This is also true if 𝒞\mathcal{C} is a σ\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.