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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

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

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

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

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

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 27th 2020

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 $f : X \to Y$ is a monomorphism, then its pushout along $X \to pt$ is a monomorphism. That is, the map $pt \to Z$ is $(-1)$-truncated where $Z$ is the cofiber of $f$. A map is $(-1)$-truncated by definition if its fibers are $(-1)$-truncated, so $\Omega(Z)$ is $-1$-truncated. In other words $Z$ is $0$-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 $\Omega(Z, g(y))$ is weakly contractible for every $y \in Y \setminus X$. We can at least see that it is discrete as follows:

Note $g : Y \to Z$ is 0-truncated, as the pushout of $X \to \pt$. So its fiber $F_y$ over any point $g(y) \in Z$ ($y \in Y$) is $0$-truncated. We may write

$\Omega(Z, g(y)) = \{g(y)\} \times_Z \{g(y)\} = \{g(y)\} \times_Z Y \times_Y \{y\} = F_y \times_Y \{y\}.$

This is a pullback of $0$-truncated objects, hence is $0$-truncated.

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