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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2010
    • (edited Oct 26th 2010)

    I started at cohesive (infinity,1)-topos a section van Kampen theorem

    In the cohesive \infty-topos itself the theorem holds trivially. The interesting part is, I think, to which extent it restricts to the concrete cohesive objects under the embedding Conc(H)HConc(\mathbf{H}) \hookrightarrow \mathbf{H}.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 26th 2010

    The page cohesive infinity-topos doesn’t exist! But your other link works.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2010

    Thanks, fixed.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2010

    I shouldn’t have gotten myself distracted by this, since I need to be doing something else, but nevertheless I did spend some time now on expanding and polishing my purported proof of the higher van Kampen theorem using cohesive \infty-topos technology here.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2010
    • (edited Oct 26th 2010)

    I worked a bit more to polish the argument that the pushout of topological spaces remains a homotopy pushout after the embedding of the spaces as 0-truncated topological \infty-groupoids.

    One can see this directly with a bit of effort, I think, but one can also fall back to the 1-categorical van Kampen theorem to see this, which is maybe noteworthy:

    for

    U 1U 2 U 1 U 2 X \array{ U_1 \cap U_2 &\to& U_1 \\ \downarrow && \downarrow \\ U_2 &\to& X }

    a cover of the topological space XX by two open subsets, pick a good open cover of XX, call the Cech nerve QXQ X and write QU iQ U_i for the Cech nerves of the restriction to those open subsets that land in U iU_i. Then one can show that the ordinary pushout

    Q(U 1)Q(U 2) Q(U 1) Q(U 2) Q(U 1) Q(U 1)Q(U 2)Q(U 2) \array{ Q(U_1) \cap Q(U_2) &\to& Q(U_1) \\ \downarrow && \downarrow \\ Q(U_2) &\to& Q(U_1) \coprod_{Q(U_1) \cap Q(U_2)} Q(U_2) }

    of simplicial presheaves over open balls is a homotopy pushout. So the question is if this is again equivalent to XX.

    Since all objects are 0-truncated, it suffices to check that the 1-truncation of the pushout is XX. It is easy to see that π 0\pi_0 agrees. To see that π 1\pi_1 of the pushout vanishes (the categorical π 1\pi_1 not the geometric one!!) pass objectwise to the geometric realization and then use the ordinary 1-van Kampen theorem.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2010

    I made it overly complicated by working in the projective model structure. Using the injective structure the whole statement becomes trivial. We can prove higher van Kampen in a single line.

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