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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2014
    • (edited Oct 19th 2014)

    I thought this would be straightforward, but now it seems I am stuck: for 𝒞\mathcal{C} an \infty-category (model category) with products and 𝒞 Δ op\mathcal{C}^{\Delta^{op}} its simplicial objects, let L Δ 1𝒞 Δ opL_{\Delta^1} \mathcal{C}^{\Delta^{op}} be the localization (Bousfield localization) at the morphisms ()×Δ 1()(-)\times \Delta^1 \to (-).

    Is GrpdL Δ 1(Grpd) Δ op\infty Grpd \simeq L_{\Delta^1} (\infty Grpd)^{\Delta^{op}}?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeOct 19th 2014
    • (edited Oct 19th 2014)

    To avoid confusion, let T nT^n be Δ n\Delta^n regarded as degreewise discrete simplicial “space”. First, let us show that the projections T nT 0T^n \to T^0 get inverted. But these are simplicial homotopy equivalences, so they become invertible if you invert the projection X×T 1XX \times T^1 \to X for every simplicial “space” XX. Thus the local objects are precisely the “constant simplicial spaces”, if I’m not mistaken.

    Left Bousfield localisation at a proper class is not known to be possible a priori, but in this case the local objects do turn out to form a reflective (,1)(\infty, 1)-subcategory, with reflector given by “geometric realisation” (i.e. codescent objects).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2014

    True, thanks. I was trying to see that from the nn-cube spaces X (Δ 1) nX^{(\Delta^1)^n} all being equivalent it follows that the nn-simplex spaces X nX_n are all equivalent. But of course what you say is the right way to do it.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 19th 2014

    I have added that remark here (in the Examples-section at cohesive (infinity,1)-topos). Feel invited to expand, if you care.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeOct 20th 2014

    Nice, thanks Zhen! Can you also characterize the objects of the slice Gpd Δ op/X\infty Gpd^{\Delta^{op}}/X localized at T 1T^1?

    • CommentRowNumber6.
    • CommentAuthorZhen Lin
    • CommentTimeOct 20th 2014

    I’m not sure. By adjunction, an object YY in the slice category is right orthogonal to X *AX *BX^* A \to X^* B if and only if XY\prod_X Y is right orthogonal to ABA \to B, so YY is right orthogonal to every Z×T 1Z×T 0Z \times T^1 \to Z \times T^0 in the slice category if and only if every X(Y Z)\prod_X (Y^Z) is right orthogonal to T 1T 0T^1 \to T^0, but I can’t really make heads or tails of that.

    The special case where XX is itself “constant” should be more straightforward. Assuming “geometric realisation” [Δ op,Grpd /X]Grpd /X[\mathbf{\Delta}^{op}, \infty \mathbf{Grpd}_{/ X}] \to \infty \mathbf{Grpd}_{/ X} continues to send simplicial homotopy equivalences to equivalences, the same analysis should work, yielding the same conclusion.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 21st 2014

    The special case where XX is itself “constant” should be more straightforward.

    Yes, internally: a map with modal codomain is modal iff it has modal fibers.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeApr 7th 2015

    Given f:YXf:Y\to X and g:ZXg:Z\to X in the slice over XX, I believe that X(f g)\prod_X (f^g) (which I assume is what you mean in #6 by XY Z\prod_X Y^Z) should be the same as Zg *f\prod_Z g^*f. So YY is right orthogonal to every Z× XT 1ZZ\times_X T^1 \to Z in the slice over XX iff g *Yg^*Y is right orthogonal to Z×T 1ZZ\times T^1 \to Z in the slice over ZZ for every g:ZXg:Z\to X. And it’s probably enough to consider the universal cases Z=X n×T nZ = X_n \times T^n