Timeline for Dold-Kan correspondence in the category of symmetric spectra
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 24, 2019 at 12:57 | comment | added | Surojit Ghosh | @DylanWilson: Can you please give me a reference for " There is an equivalence between the homotopy theory of filtered objects and simplicial objects in spectra"? | |
| Jun 23, 2019 at 21:32 | comment | added | Surojit Ghosh | @Dylan Wilson: Sorry for the unclear question. But I really mean that the category of simplicial objects in symmetric spectra not the EM- modules. | |
| Jun 23, 2019 at 18:32 | comment | added | Dylan Wilson | I have no idea, I didn’t really understand the original post- I was just responding to the “My precise question is” where the OP specifies starting with simplicial objects in spectra, not simplicial objects in modules over an EM spectrum. Maybe they can clarify? | |
| Jun 23, 2019 at 17:01 | comment | added | David White | @DylanWilson: I think he is working with modules over an EM-ring spectrum, unless I misinterpreted the question again. Why else lead with the Dold-Kan stuff? | |
| Jun 23, 2019 at 15:54 | comment | added | Dylan Wilson | (Here I assume that by “chain complex of spectra” you mean a sequence of spectra with maps d between them such that d^2 is null, or maybe even you want to remember the null homotopy- that’s fine too, but it’ll still be a loss of too much info) | |
| Jun 23, 2019 at 15:49 | comment | added | Dylan Wilson | I think the answer to your question should be “no”. There is an equivalence between the homotopy theory of filtered objects and simplicial objects in spectra. If you take the associated graded of a filtered object with its “d1 differential” then you get a chain complex in spectra. If you were working with modules over EM-ring-spectra then that’s not so bad, but over the sphere spectrum that’s a huge loss of information. | |
| Jun 23, 2019 at 13:38 | comment | added | David White | In that case, you should look at the paper of Richter and Shipley: math.uni-hamburg.de/home/richter/commhzrev.pdf. They deal with spectra in chain complexes (which should be the same as chain complexes in spectra, since both are diagram categories). Symmetric spectra is already simplicial and stable, so I think the answer to your question should be "yes" | |
| Jun 23, 2019 at 13:13 | comment | added | Surojit Ghosh | @DavidWhite: My precise question is: Is there any categorical equivalence or Quillen equivalence between the category of simplicial objects in symmetric spectra, $s(Sp^\Sigma)$, and category of chain complexes in $Sp^\Sigma$? | |
| Jun 23, 2019 at 12:38 | comment | added | David White | I had trouble parsing the question, so I just answered both ways I could interpret it (i.e. Q1: can you do Dold-Kan in symmetric spectra, and Q2: can you do Dold-Kan with respect to M instead of N) | |
| Jun 23, 2019 at 12:37 | comment | added | Denis Nardin | Actually, I misspoke: I was thinking of the equivalence between simplicial spectra and filtered spectra (which Lurie calls the "∞-categorical Dold-Kan correspondence"), not of $HR$-modules with the derived category of $R$ | |
| Jun 23, 2019 at 12:35 | history | answered | David White | CC BY-SA 4.0 |