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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 18th 2013
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   Author: Urs
   Format: MarkdownItexI wanted to collect some of the stuff recently added to a bunch of chromatic entries in a way that forms an at least semi-coherent story, so I made an entry * _[[J-homomorphism and chromatic homotopy]]_. This is built mostly from copy-and-pasting stuff that I had added to dedicated entries, equipped with a bit of glue to make it stick together and form a story. (Special thanks to Marc Hoyois for general discussion and in particular for working on the text on the [[Lurie spectral sequence]].) I want to further fine-tune this. But not tonight.

   I wanted to collect some of the stuff recently added to a bunch of chromatic entries in a way that forms an at least semi-coherent story, so I made an entry

   • J-homomorphism and chromatic homotopy.

   This is built mostly from copy-and-pasting stuff that I had added to dedicated entries, equipped with a bit of glue to make it stick together and form a story.

   (Special thanks to Marc Hoyois for general discussion and in particular for working on the text on the Lurie spectral sequence.)

   I want to further fine-tune this. But not tonight.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 3rd 2013
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   Author: Urs
   Format: MarkdownItexAm brushing up the section _[Canonical cosimplicial resolution of E-infinity algebra](http://ncatlab.org/nlab/show/J-homomorphism+and+chromatic+homotopy#CanonicalCosimplicialResolutionOfEInfinityAlgebras)_. It used to say there that it is the $E_2$-page of the [[spectral sequence of a simplicial stable homotopy type]] which is given by the Moore complexes of the underlying simplicial objects of homotopy groups. I originally took this from Dylan Wilson's [prop. 1.2.3](http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-03-21-Dylan-Wilson-ANSS.pdf) But it's instead the $E_1$-page, isn't it.

   Am brushing up the section Canonical cosimplicial resolution of E-infinity algebra.

   It used to say there that it is the $E_2$-page of the spectral sequence of a simplicial stable homotopy type which is given by the Moore complexes of the underlying simplicial objects of homotopy groups. I originally took this from Dylan Wilson’s prop. 1.2.3 But it’s instead the $E_1$-page, isn’t it.

3. • CommentRowNumber3.
   • CommentAuthorDylan Wilson
   • CommentTimeDec 3rd 2013
   • (edited Dec 3rd 2013)
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   Author: Dylan Wilson
   Format: MarkdownItexWhat you have should be the E_2-page. The E_1 page is pi_\astTot(cosk_\astY) with a differential coming from alternating sum. Taking homology gives the E_2 page, as stated. Also it probably says this elsewhere, but convergence of said spectral sequence is delicate in this "cosimplicial" case as opposed to the simplicial case.

   What you have should be the E_2-page. The E_1 page is pi_\astTot(cosk_\astY) with a differential coming from alternating sum. Taking homology gives the E_2 page, as stated.

   Also it probably says this elsewhere, but convergence of said spectral sequence is delicate in this “cosimplicial” case as opposed to the simplicial case.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 4th 2013
   • (edited Dec 4th 2013)
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   Author: Urs
   Format: MarkdownItexOh, sure. I am being stupid, ignored the $H_p$. I plan to write up a more detailed discussion of the cosimplicial spectral sequence by next Tuesday. So maybe I will come back to you again with more stupid questions.

   Oh, sure. I am being stupid, ignored the $H_p$.

   I plan to write up a more detailed discussion of the cosimplicial spectral sequence by next Tuesday. So maybe I will come back to you again with more stupid questions.