Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
am giving this an entry of its own, split off from motivic homotopy theory.
Nothing much here yet, just a bare minimum so far
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?
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?
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.
That’s just the homotopy fiber at the basepoint. Why is that sufficient?
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.
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.
1 to 7 of 7