Timeline for Monoidality of truncation of spectra
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 19, 2016 at 1:17 | comment | added | Dmitry Vaintrob | @Tyler, Thanks! I meant a reference for the [-j:0] truncations preserving symmetric monoidal structure | |
| Dec 18, 2016 at 21:48 | answer | added | David White | timeline score: 2 | |
| Dec 18, 2016 at 18:35 | comment | added | Tyler Lawson | @Dmitry I don't know an automatic reference. For $[i,j]$ to automatically preserve $E_n$-ring structures, the interval has to include 0 (otherwise the unit can't map in), and 0 has to be one of the endpoints (otherwise you could consider an example where you have a unit in degree 1 with inverse in degree -1, which can't be preserved by truncation). So that just leaves us to exclude the cases $[i,0]$. For these the natural maps relating it to $R$ tend to be in the wrong direction (you take the quotient by the elements in degree less than 0, for example) to preserve multiplication. | |
| Dec 17, 2016 at 19:01 | comment | added | Dmitry Vaintrob | @DylanWilson, sorry, I should have put up a co-trigger warning | |
| Dec 17, 2016 at 19:01 | comment | added | Dmitry Vaintrob | @TylerLawson, thanks! Do you have a reference? | |
| Dec 17, 2016 at 18:46 | comment | added | Dylan Wilson | Today I learned that cohomological notation with lower indices freaks me out. | |
| Dec 17, 2016 at 18:27 | comment | added | Tyler Lawson | I believe that only the truncations $X_{[0:-j]}$ for $j \geq 0$ in your notation are guaranteed to preserve $E_n$ ring spectra. | |
| Dec 17, 2016 at 15:04 | history | edited | Denis Nardin | CC BY-SA 3.0 |
Monoidality, not monoidicity
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| Dec 17, 2016 at 15:00 | history | edited | Dmitry Vaintrob | CC BY-SA 3.0 |
minor edits
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| Dec 17, 2016 at 14:51 | comment | added | Denis Nardin | I think it is lax symmetric monoidal (so it takes $E_n$-algebras to $E_n$-algebras), since it is a symmetric monoidal localization. | |
| Dec 17, 2016 at 14:37 | history | edited | Dmitry Vaintrob | CC BY-SA 3.0 |
cohomological
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| Dec 17, 2016 at 14:28 | comment | added | Dmitry Vaintrob | Did I get my arrows backwards? The map $X^i\wedge \Sigma^\infty BS_i\to X$ should canonically preserve the connective part | |
| Dec 17, 2016 at 14:26 | comment | added | Charles Rezk | Is $X\mapsto X_{\leq 0}$ really symmetric monoidal? | |
| Dec 17, 2016 at 14:16 | history | asked | Dmitry Vaintrob | CC BY-SA 3.0 |