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At some point I had made up the extra axiom/terminology saying that an object $\mathbb{A}^1$ in a cohesive $\infty$-topos “exhibits the cohesion” if the shape modality is equivalent to $\mathbb{A}^1$-localization. Now I was talking about that assumption with Mike and noticed that this didn’t have a reflection on the $n$Lab yet.
So now I have added, for the record, the definition here at “cohesive oo-topos” and cross-linked with the existing discussion at “continuum”.
Is there any analogous property for the sharp modality?
None such ever occurred to me. But I have not looked for it either.
Which cohesive $\infty$-toposes have such an $\mathbb{A}^1$? Smooth∞Grpd and ETop∞Grpd, yes. What of the super- ones?
Maybe Charles Rezk’s global spaces doesn’t.
What’s the easiest way to show that no such $\mathbb{A}^1$ is present?
Right, so for $SmoothSuper\infty Grpd$ regarded as cohesive over $\infty Grpd$ there is not just one $\mathbb{A}^1$, but the whole family of super-lines $\{\mathbb{R}^{1|N}\}$ for $N \in \mathbb{N}$. But, as we discussed elsewhere, $SmoothSuper\infty Grpd$ is more naturally regaded as being cohesive over $Super \infty Grpd$, and as such it is again $\mathbb{R}^1 = \mathbb{R}^{1|0}$ which exhibits the cohesion! The analogous statement holds for $FormalSmooth\infty Grpd$ and for $FormalSmoothSuper\infty Grpd$ etc.
Indeed, just a few hours back I was discussing with Mike, saying that I haven’t really checked, but that I suspect the global equivariant cohesion may not have an object exhibiting cohesion, and that maybe the presence of such an object is the demarkation line between cohesion to which the evident geometric intuition applies, and the more surprising or exotic cohesion such as in global equivariance.
But I haven’t checked, and I haven’t thought of a general criterion. Maybe there is an $\mathbb{A}^1$ for the global equivariant cohesion after all. I don’t know. Would be good if this could be decided. Maybe somebody here has an idea?!
If I understand what you are saying, “shape” in the global equivariant theory is localization with respect to the collection of maps $\{ \mathbb{B}G\to * \}$, ranging over all groups $G$.
@Charles, the question that we are wondering about is whether that is equivalent to localization at a class of maps of the form
$(-) \times \mathbb{A}^1 \to (-)$for some object $\mathbb{A}^1$ in the global equivariant homotopy theory.
A general question that would be good to have answers to is this:
suppose some object $\mathbb{A}^1$ exhibits the cohesion in that $\Pi \simeq L_{\mathbb{A}^1}$, are there some good conditions to imply that then $\Pi$ preserves homotopy pullback over discrete objects?
Or: that the $\{\mathbb{A}^n\}_{n \in \mathbb{N}}$ form a site of definition?
If the localization at some class of maps happens to be an exponential ideal, which the shape modality always is, then it is automatically also the localization at the closure of that class under products (i.e. the “internal localization”). So Charles’ statement implies that the global equivariant shape is also the localization at the maps $(-) \times BG \to (-)$. It’s not surprising to me if there is no one object that can play that role, since the point of global equivariant theory is to look at all groups at once. How important is it for your applications, Urs, that the shape is a localization at a single object rather than a family of them?
Well, in equivariant motivic homotopy theory, one also localizes at all G-affine bundles, not just A^1. I don’t think this changes much of the formal theory.
If the localization at some class of maps happens to be an exponential ideal, which the shape modality always is, then it is automatically also the localization at the closure of that class under products (i.e. the “internal localization”). So Charles’ statement implies that the global equivariant shape is also the localization at the maps $(-) \times BG \to (-)$.
Ah, right. Thanks.
How important is it for your applications, Urs, that the shape is a localization at a single object rather than a family of them?
I haven’t done a whole lot yet with the $\mathbb{A}^1$-objects in cohesion apart from noticing that they exist in the standard models.
In the smooth model, one key aspect is that the simplices built from the $\mathbb{A}^1$-object induce an internal singular simplicial complex construction which models $\Pi$.
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