Timeline for "Universal" triangulated category
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11 events
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| Aug 9, 2020 at 20:27 | history | became hot network question | |||
| Aug 9, 2020 at 16:34 | vote | accept | curious math guy | ||
| Aug 9, 2020 at 15:48 | history | edited | David White | CC BY-SA 4.0 |
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| Aug 9, 2020 at 15:38 | answer | added | David White | timeline score: 14 | |
| Aug 9, 2020 at 14:37 | comment | added | Harry Gindi | @DylanWilson The example I'm thinking of here is where C is something that is already abelian or at least additive, so I think the point is that CMG is implicitly thinking of the construction Ho Stab Ch^+(C) (identifying Ch^+(C) with sC, in which case this does have a universal property I think. | |
| Aug 9, 2020 at 13:28 | comment | added | Dylan Wilson | more stuff: if you'd like to take an ordinary category $C$ and then view $sC$ as simplicially enriched, I think you need $C$ to be complete or cocomplete (otherwise how do you define the simplicial structure on hom-objects?). Also, another issue you might run into is that the construction 'sC' isn't very universal, as far as I know... for specific examples of C it ends up having a universal property as a sifted cocompletion of a certain subcategory of C, but I don't know a good invariant description in general | |
| Aug 9, 2020 at 12:56 | history | edited | curious math guy | CC BY-SA 4.0 |
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| Aug 9, 2020 at 12:40 | history | edited | curious math guy | CC BY-SA 4.0 |
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| Aug 9, 2020 at 12:36 | comment | added | curious math guy | @DylanWilson Thank you very much! I'll change my question accordingly. | |
| Aug 9, 2020 at 12:33 | comment | added | Dylan Wilson | I think there's some confusions here that are worth sorting through. (1) 'associated simplicial category' would usually mean C back again, but you seem to use it later to mean the category of simplicial objects in C, with its simplicial structure? (2) the htpy category of a stable $\infty$-category has a canonical triangulated structure, not just any old $\infty$-category, (3) the homotopy category of sA, for A abelian, is not $D(A)$, or triangulated, but rather $D_{\ge 0}(A)$... I think there are some other issues, but that's a start | |
| Aug 9, 2020 at 12:23 | history | asked | curious math guy | CC BY-SA 4.0 |