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briefly added to infinity-group of units the statement that sending $E_\infty$-rings to their $\infty$-group of units is a right adjoint, due to ABGHR08.
Added the same also to abelian infinity-group.
made ∞-group ∞-ring a redirect to ∞-group of units, to be eventually split off in a stand-alone entry…
Have added a subsection Definition – Augmented definition with some items from Sagave’s article.
I didn’t realize earlier from just reading his introduction that in fact by his lemmas 2.12 and 3.16 there is a map from the ordinary $\infty$-group of units to the “graded” $\infty$-group of units
$gl_1(E) \to gl_1^J(E) \to \mathbb{S}$which of course means that also the ordinary $gl_1(E)$ is caonically $\mathbb{S}$-graded.
This seems to be noteworthy (and so I made a note in the entry at the above link), for it is the ordinary $gl_1(E)$ that appears in the $\infty$-adjunction with $\mathbb{S}[-]$ and notably in the definition of the twists of $E$-cohomology. So it is important that already the ordinary $\infty$-group of units is canonically $\mathbb{S}$-graded.
Right and left adjoints seemed mixed up in two places, so I corrected these. Pretty sure I got it right.
Yes, you are right. Thanks for catching that!
added to infinity-group of units, to group of units and to the corresponding section at affine line the comment that
$GL_1(R) \simeq [Spec R, \mathbb{G}_m]$added to infinity-group of units a quick paragraph Properties – Cohomology and logarithm
We might call S[A] the spring ∞-group ∞-ring of A over the sphere spectrum.
What’s ’spring’?
Oh, maybe it’s spam. I’ve taken out a stray ’spring’ before.
Quick search, TCFT
This is the result of spring Cos04 reformulated and generalized according to ClassTFT, theorem 4.2.14.
Have we had this kind of attack before?
It looks like Urs in February made the change ! hardly ‘spring’! It is not clear what it should be.
‘string’?
…. my thought also but the sentence does not parse properly even then. I also thought a mixture of ’SPectrum’ and ’RING’, but that does not work either.
The sentence may not parse in ordinary language, but it looks like good Urs-language to me. Compare string 2-group.
Sorry for causing this!
Here is the unbelievable truth:
“spring” is German for “jump”. In the process of editing entries I frequently insert “spring” in places that I need to jump back to after doing some edits elsewhere: then I just Ctrl-F for “spring” and am back.
After done with editing, the “spring” markers are supposed to be removed of course. Here I had forgotten to remove it. Done now. Sorry again.
Ah! I do the same thing with ‘%%’ (ultimately derived from TeX).
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