# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 9th 2013

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 9th 2013

made ∞-group ∞-ring a redirect to ∞-group of units, to be eventually split off in a stand-alone entry…

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 20th 2013

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.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeSep 18th 2013

Right and left adjoints seemed mixed up in two places, so I corrected these. Pretty sure I got it right.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 18th 2013

Yes, you are right. Thanks for catching that!

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 7th 2013

added to infinity-group of units, to group of units and to the corresponding section at affine line the comment that

$GL_1(R) \simeq [Spec R, \mathbb{G}_m]$
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 26th 2014
• (edited Mar 26th 2014)
• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeAug 22nd 2014

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?

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeAug 22nd 2014

It looks like Urs in February made the change ! hardly ‘spring’! It is not clear what it should be.

• CommentRowNumber10.
• CommentAuthorTobyBartels
• CommentTimeAug 23rd 2014

‘string’?

• CommentRowNumber11.
• CommentAuthorTim_Porter
• CommentTimeAug 23rd 2014
• (edited Aug 23rd 2014)

…. 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.

• CommentRowNumber12.
• CommentAuthorTobyBartels
• CommentTimeAug 23rd 2014

The sentence may not parse in ordinary language, but it looks like good Urs-language to me. Compare string 2-group.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeAug 23rd 2014
• (edited Aug 23rd 2014)

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.

• CommentRowNumber14.
• CommentAuthorTobyBartels
• CommentTimeAug 24th 2014

Ah! I do the same thing with ‘%%’ (ultimately derived from TeX).

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeSep 4th 2020

added pointer to the published version of ABGHR 08:

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeSep 5th 2020

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeSep 5th 2020
• (edited Sep 5th 2020)

The entry claims that the notion of the group of units of a ring spectrum goes back to

• Peter May, $E_\infty$ ring spaces and $E_\infty$ ring spectra Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave (pdf)

But where in that text does it appear? I don’t see it.