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nLab > Latest Changes: infinity-group of units

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2013

    briefly added to infinity-group of units the statement that sending E ∞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)→gl 1 J(E)→𝕊 gl_1(E) \to gl_1^J(E) \to \mathbb{S}

    which of course means that also the ordinary gl 1(E)gl_1(E) is caonically 𝕊\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)gl_1(E) that appears in the ∞\infty-adjunction with 𝕊[−]\mathbb{S}[-] and notably in the definition of the twists of EE-cohomology. So it is important that already the ordinary ∞\infty-group of units is canonically 𝕊\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)≃[SpecR,𝔾 m] 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:

    diff, v40, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2020

    added pointer to

    diff, v41, current

    • 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 ∞E_\infty ring spaces and E ∞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.)

    diff, v41, current

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