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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 9th 2012
   • (edited Feb 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexPrompted by an email question that I received, I noted that the entry _[[infinite loop space]]_ was (and is) in bad shape. I have now added some references to it and a sentence including the word "$E_\infty$". Maybe I find time to do a bit more...

   Prompted by an email question that I received, I noted that the entry infinite loop space was (and is) in bad shape.

   I have now added some references to it and a sentence including the word “$E_\infty$”. Maybe I find time to do a bit more…

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 9th 2012
   • (edited Feb 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItextime for a table. I am experimenting with this here: | | [[(∞,1)-operad]] | [[∞-algebra over an (∞,1)-operad]] | groupal version | in [[Top]] | generally | |--|----------------|--------------------------------|----|--------|------| | | [[A-∞ operad]] | [[A-∞ algebra]] | [[∞-group]] | [[loop space]] | [[loop space object]] | | | [[E-k operad]] | [[E-k algebra]] | [[k-monoidal ∞-group]] | [[iterated loop space]] | [[iterated loop space object]] | | | [[E-∞ operad]] | [[E-∞ algebra]] | [[abelian ∞-group]] | [[infinite loop space]] | [[infinite loop space object]] | | | | | | $\simeq$ [[connective spectrum]] | $\simeq$ [[connective spectrum object]] | | **[[stabilization]]** | | | | [[spectrum]] | [[spectrum object]] | A version of this is currently at [[infinite loop space]], but I think I'll make it an include-entry now

   time for a table. I am experimenting with this here:

   	(∞,1)-operad	∞-algebra over an (∞,1)-operad	groupal version	in Top	generally
   	A-∞ operad	A-∞ algebra	∞-group	loop space	loop space object
   	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
   	E-∞ operad	E-∞ algebra	abelian ∞-group	infinite loop space	infinite loop space object
   				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
   stabilization				spectrum	spectrum object

   A version of this is currently at infinite loop space, but I think I’ll make it an include-entry now

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 18th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded [ISBN:9780691082066](https://press.princeton.edu/books/paperback/9780691082066/infinite-loop-spaces-am-90-volume-90) and [doi:10.1515/9781400821259](https://doi.org/10.1515/9781400821259) to: * [[John Adams]], _Infinite loop spaces_, Hermann Weyl lectures at IAS, Princeton University Press (1978) ([ISBN:9780691082066](https://press.princeton.edu/books/paperback/9780691082066/infinite-loop-spaces-am-90-volume-90), [doi:10.1515/9781400821259](https://doi.org/10.1515/9781400821259)) <a href="https://ncatlab.org/nlab/revision/diff/infinite+loop+space/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinite+loop+space/9">v9</a>, <a href="https://ncatlab.org/nlab/show/infinite+loop+space">current</a>

   added ISBN:9780691082066 and doi:10.1515/9781400821259 to:

   • John Adams, Infinite loop spaces, Hermann Weyl lectures at IAS, Princeton University Press (1978) (ISBN:9780691082066, doi:10.1515/9781400821259)

   diff, v9, current

4. • CommentRowNumber4.
   • CommentAuthorHurkyl
   • CommentTimeJul 16th 2022
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexA point that confused me for a long time is that the literature about infinite loop spaces isn't always clear whether or not they are intended to come equipped with a distinguished choice of deloopings. As I understand, the equivalences with notions of connective spectra and E∞ groups are only for when you equip infinite loop spaces with a distinguished choice of deloopings, so I've added language to that effect. Is there standard language for expressing this? My choice of language is to say "infinite loop space with a choice of deloopings" which feels awkward. <a href="https://ncatlab.org/nlab/revision/diff/infinite+loop+space/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinite+loop+space/12">v12</a>, <a href="https://ncatlab.org/nlab/show/infinite+loop+space">current</a>

   A point that confused me for a long time is that the literature about infinite loop spaces isn’t always clear whether or not they are intended to come equipped with a distinguished choice of deloopings.

   As I understand, the equivalences with notions of connective spectra and E∞ groups are only for when you equip infinite loop spaces with a distinguished choice of deloopings, so I’ve added language to that effect.

   Is there standard language for expressing this? My choice of language is to say “infinite loop space with a choice of deloopings” which feels awkward.

   diff, v12, current

5. • CommentRowNumber5.
   • CommentAuthorDoronGrossmanNaples
   • CommentTimeDec 29th 2022
   • PermaLink
   Author: DoronGrossmanNaples
   Format: MarkdownItexCorrected the reference to HA. It previously pointed to the wrong section; now it points to the specific theorem containing Lurie's formulation of the loop-space principle. <a href="https://ncatlab.org/nlab/revision/diff/infinite+loop+space/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinite+loop+space/13">v13</a>, <a href="https://ncatlab.org/nlab/show/infinite+loop+space">current</a>

   Corrected the reference to HA. It previously pointed to the wrong section; now it points to the specific theorem containing Lurie’s formulation of the loop-space principle.

   diff, v13, current