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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.
Late night thought:
Do Tate spheres represent configuration spaces of points?
For consider, in the smooth category (i.e. in the sheaf topos over the site of smooth manifolds), morphisms from Cartesian space to its Tate sphere:
$\mathbb{R}^n \longrightarrow \mathbb{R}^n/ (\mathbb{R}^n \setminus \{0\})$We get such morphisms from, in particular, a choice of point $x \in \mathbb{R}^n$, namely induced from any affine map $\mathbb{R}^n \to \mathbb{R}^n$ which takes the origin to $x$. This is such that any other point in $\mathbb{R}^n$ goes to the base point of the Tate sphere, while the germ around $x$ maps to identically wrap around the Tate sphere.
Now, sheafifying this construction, we see that general maps $\mathbb{R}^n \longrightarrow \mathbb{R}^n/ (\mathbb{R}^n \setminus \{0\})$ include those given by any configuration of points in $\mathbb{R}^n$, sending the germ around any point in the configuration to identically wrap the Tate sphere and sending all points not in the configuration to the base point.
It feels like this construction should exhaust the set of such maps, though I don’t see right now how to prove this. But if true, then the internal hom $Maps\big(\mathbb{R}^n, \mathbb{R}^n/ (\mathbb{R}^n \setminus \{0\}\big)$ should be close to being the unordered configuration space of points in $\mathbb{R}^n$, with its canonical smooth structure. (Or a bit fatter than that, including the choice of a germ of frames around each point.)
Just thinking out loud here. Is anything like this discussed in the literature? (The algebraic geometers might substitute “Hilbert scheme” for “configuration space”, throughout.)
Linked to Tate twist, both in the ’Related concepts’ section and in the introduction. Added some brief remarks on the relationship.
Is there a reason by the way to name this page ’Tate sphere’ rather than ’Tate circle’ this point? If we keep the name ’Tate sphere’, we should add ’Tate circle’ as a redirect I think.
The Tate circle is $\mathbb{A}^1 \setminus \{0\}$.
The Tate sphere is $\mathbb{A}^1/(\mathbb{A}^1 \setminus \{0\})$.
(E.g. Voevodsky 07 p. 3, p. 13).
(Some people do say “Tate sphere” for the Tate circle, but not conversely, as far as I am aware. Generally, it’s common to call a circle a “1-sphere”, but nobody calls the sphere a “2-circle”.)
Yes, after I wrote the above I reflected that Tate circle would be confusing. I’m not massively keen on Tate sphere, but I suppose the terminology comes from the fact that it is $\mathbb{A}^1$-weakly equivalent to $\mathbb{P}^1$, which over $\mathbb{C}$ is a sphere of course. A bit tenuous in general though, especially for this page where it is considered more generally than the setting of $\mathbb{A}^1$-homotopy theory.
And what’s the relation to Tate twists?
I tried to roughly explain this in the entry. When forming the stable motivic homotopy category, we invert smashing with $\mathbb{P}^{1}$. The difference between doing this and inverting smashing with $S^{1}$ is exactly the ability to represent the Tate twist part of motivic cohomology theories. If you’re looking for something precise, see for example Theorem 10.2 in Mazza, Voevodsky, Weibel: $H^{p,q}_{L}(X, \mathbb{Z} / n)$, the étale motivic cohomology of $X / k$ with coefficients in $\mathbb{Z} / n$ where $n$ is prime to the characteristic of $k$, is isomorphic $H^{p}_{et}(X, \mathbb{Z}/n(q))$, the étale cohomology of $X$ with coefficients in the $q$-th Tate twist of $\mathbb{Z}/n$. And to represent the $q$ part of motivic cohomology, i.e. the Tate twist part of étale cohomology with these coefficients, is exactly what inverting the Tate sphere (or in other words $\mathbb{G}_{m}$ as well as $\mathbb{S}^{1}$) rather than just $S^{1}$ allows.
I am not convinced of the deep significance of the description $\mathbb{A}^{1} / \mathbb{G}_{m}$ motivically really. It is $\mathbb{P}^{1}$ that is really important, and the description $S^{1} \wedge \mathbb{G}_{m}$ is also important in $\mathbb{A}^{1}$-homotopy theory.
As I wrote in the entry, it is essentially the same story for pure motives, taking again the fact that the Tate sphere is $\mathbb{P}^{1}$ as primary.
I tried to roughly explain this in the entry.
In which entry? At Tate sphere is just says
The Tate sphere gives rise to Tate twists
The sentences that follow just say that the Tate sphere is inverted in motivic homotopy theory.
That’s why I was asking.
If you’re looking for something precise
Since we are in the Idea section, something imprecise would be okay, possibly even desireable, but it should be contentful.
The paragraph starting with
In formal constructions of categories of motives, one typically ‘inverts the Tate sphere’
is reasonable at this point, but the two paragraphs that follow would seem to better fit in an entry like motivic homotopy theory. To keep them here would mean to effectively merge the entry Tate sphere back into the general discussion of motivic homotopy theory from which I had split it off for focus. I suggest you cut-and-paste this material to where it fits better.
The idea was to explain that the inverting of the Tate sphere is not just about representing motivic Tate twists, but ordinary ones in étale cohomology as well. But I’ve removed the two paragraphs now and added them to étale cohomology.
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