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.

(Maybe I had once added that reference myself, but i forget.)

]]>added pointer to

- Peter May, Johann Sigurdsson, Section 22.2 of:
*Parametrized Homotopy Theory*, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)

added pointer to the published version of ABGHR 08:

- Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk,
*Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory*, Journal of Topology, Volume7, Issue 4, December 2014 (arXiv:1403.4320, arXiv:10.1112/jtopol/jtu009)

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

]]>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.

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

]]>…. 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.

]]>‘string’?

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

]]>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?

]]>added to *infinity-group of units* a quick paragraph *Properties – Cohomology and logarithm*

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

Yes, you are right. Thanks for catching that!

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

]]>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.

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

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*.