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    • CommentRowNumber1.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 23rd 2013

    Fixed a couple incorrect statements at hypercomplete (infinity,1)-topos:

    • Remark 1 claimed that having enough points in the 1-topos sense implies having enough points in the ∞-sense. I replaced it with a counterexample from HTT.
    • In Proposition 1 I replaced “finite homotopy dimension” by “locally of homotopy dimension n\leq n”.
    • CommentRowNumber2.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 23rd 2013

    Here’s a simple counterexample to “finite homotopy dimension implies hypercompleteness”:

    Let H\mathbf{H} be the big (∞,1)-topos on the site TopTop of topological spaces. Then H\mathbf{H} has homotopy dimension 00 but is not hypercomplete because it contains non-hypercomplete (∞,1)-topoi as essential retracts of its slices. More precisely, if XX is a topological space, the embedding i:Open(X)Top /Xi\colon Open(X)\to Top_{/X} is continuous and cocontinuous, so that we have an adjunction i !i *i *i_!\vdash i^*\vdash i_* with i !i_! fully faithful. In that situation, i *i^* preserves nn-truncated morphisms, hence i !i_! preserves \infty-connective morphisms, hence i *i^* preserves hypercomplete objects, hence i !i_! preserves non-hypercomplete objects. Thus, for appropriate XX, H /X\mathbf{H}_{/X} and hence H\mathbf{H} are not hypercomplete.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2013
    • (edited Nov 24th 2013)

    Remark 1 claimed that having enough points in the 1-topos sense implies having enough points in the ∞-sense. I replaced it with a counterexample from HTT.

    Thanks for looking into this. But wait, the argument that you removed had just another “hypercomplete” implicit/missing.

    The argument said that in the Jardine model structure it is true that when the underlying 1-topos has enough points, then the Jardine weak equivalences are equivalently the stalkwise weak equivalences. Since the Jardine model structure presents the hypercomplete \infty-topos over the site, it does follow that this has enough points when the 1-topos does.

    • CommentRowNumber4.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 24th 2013

    I added the remark that the hypercompletion has enough points.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2013

    Okay, thanks.

    (yeah, that darn hypercompletion subtlety… ;-)

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