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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2020
    • (edited Jun 14th 2020)

    am giving this an entry of its own, split off from motivic homotopy theory.

    Nothing much here yet, just a bare minimum so far

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2020
    • (edited Jun 14th 2020)

    Incidentally, what is the right way to say “complement” V{x}V \setminus \{x\} more generally in a topos, for a general pointed object VV?

    I suppose: the union of all subobjects UVU \hookrightarrow V such that we have a Cartesian square

    * x U V \array{ \varnothing &\longrightarrow& \ast \\ \big\downarrow && \big\downarrow{\mathrlap{{}^x}} \\ U &\hookrightarrow& V }

    Not a big deal, I suppose, but does anyone discuss Tate spheres in this generality?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2020

    I am being dumb, please help:

    Given a monomorphism between 0-truncated objects in an \infty-topos, what’s the abstract argument (not via simplicial presheaves) that its homotopy cofiber is also 0-truncated?

    • CommentRowNumber4.
    • CommentAuthorjalfy
    • CommentTimeJun 27th 2020
    • (edited Jun 27th 2020)

    If f:XYf : X \to Y is a monomorphism, then its pushout along XptX \to pt is a monomorphism. That is, the map ptZpt \to Z is (1)(-1)-truncated where ZZ is the cofiber of ff. A map is (1)(-1)-truncated by definition if its fibers are (1)(-1)-truncated, so Ω(Z)\Omega(Z) is 1-1-truncated. In other words ZZ is 00-truncated.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2020

    That’s just the homotopy fiber at the basepoint. Why is that sufficient?

    • CommentRowNumber6.
    • CommentAuthorjalfy
    • CommentTimeJun 27th 2020
    • (edited Jun 27th 2020)

    Sorry, was thinking of the pointed setting.

    So we have to show Ω(Z,g(y))\Omega(Z, g(y)) is weakly contractible for every yYXy \in Y \setminus X. We can at least see that it is discrete as follows:

    Note g:YZg : Y \to Z is 0-truncated, as the pushout of XptX \to \pt. So its fiber F yF_y over any point g(y)Zg(y) \in Z (yYy \in Y) is 00-truncated. We may write

    Ω(Z,g(y))={g(y)}× Z{g(y)}={g(y)}× ZY× Y{y}=F y× Y{y}.\Omega(Z, g(y)) = \{g(y)\} \times_Z \{g(y)\} = \{g(y)\} \times_Z Y \times_Y \{y\} = F_y \times_Y \{y\}.

    This is a pullback of 00-truncated objects, hence is 00-truncated.

    Somehow I also don’t see a clean argument that it is (1)(-1)-truncated though.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2020

    Thanks. Notice that I really do need this for sheaves, where we cannot argue with global points.

    Specifically in view of the topic of this thread I am looking at a class of examples where the pushout has only two global points in the first place, hence is far from being a concrete sheaf.

    Incidentally, it’s trivial to see the statement in terms of prsentation by an injective model structure, using that the (-1)-truncated morphism is then represented by a monomorphism of presheaves, which is hence a cofibration of simplicial presheaves.

    But it feels like this statement should have a quick intrinsic argument.

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