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

    It should be the case that CRingop is coreflective in (CRingΔop)op, the coreflection being 0-truncation of simplical rings

    CRingopτ0(CRingΔop)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 is a category of finitary algebraic structures (= locally finitely sifted-presentable) then we have

    π0ΔΓ:𝒞[Δop,𝒞]

    with π0 preserving finite products and Δ fully faithful. This is also true if 𝒞 is a σ-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.