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It seems that the story at super line 2-bundle finds its natural interpretation actually in the augmented infinity-groups of units of Sagave in
From Sagave’s theorem 1.2 It feels like the following should be true, but I need to properly check:
Assertion. The geometric realization of the group super 2-stack of super line 2-bundles sits inside what Sagave’s $bgl_1^\ast$-operation yields when applied to the complex K-theory spectrum:
${\vert 2\mathbf{sLine}\vert} \subset bgl_1^\ast(KU) \,.$Certainly the homotopy groups on both sides match. But I need to formally convince myself that there is a map that exhibits the homotopy equivalence.
Ah, yes, it works. It follows from example 4.10 in Sagave’s article.
(More later, am on the road…)
There must be a super-duper higher parallel of your assertion
${\vert n \mathbf{???Line}\vert} \simeq bgl_1^\ast(???) \,.$that we could then name the
Schreiber higher super-geometry conjecture.
I have now finally double checked: the statement is indeed that there is a homotopy equivalence between the geometric realization of the super 2-stack of super line 2-bundles and the 4-truncation of Steffen Sagave’s non-connected delooping of the $\infty$-group of units of $KU$:
${\vert \mathbf{Super2Line}\vert} \simeq bgl_1^\ast(KU) \langle 0,...,4\rangle \,.$I put that statement into the entry here, but of course more details should eventually go there.
re3:
that would be fun :-) But maybe first I need to understand what the systematics is behind that truncation. There seem to be two possibilities for speculation:
maybe in higher-truncated supersymmetry in the Kapranov sense, we can capture all of $bgl_1^\ast(KU)$ by super line 2-bundles;
or else maybe there is a good sense in which the low degree-twists which are captures by super line 2-bundles are already the “good” ones that one should focus on in some contexts.
Not sure…
And could passing towards 3-bundles change the spectrum to $tmf$?
maybe there is a good sense in which the low degree-twists which are captures by super line 2-bundles are already the “good” ones that one should focus on in some contexts.
Following the Tao of things, won’t we be led, whether we like it or not, to the fully untruncated truth of stable cohesion?
Did you see Mike’s Cafe comment which includes
Homotopy type theory seems to be telling us that constructive homological algebra is also much easier to do over the sphere spectrum than over the integers.
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