Timeline for Homotopy factorization of morphisms of chain complexes
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 12, 2014 at 13:47 | comment | added | domenico fiorenza | Dear Eric, the short note Fosco and me were needing the answer to the above question is now ready. We are planning to upload that on the arXiv soon, but a hopefully final version of it is already available here: nforum.mathforge.org/discussion/6157/… | |
| Jun 4, 2014 at 19:17 | comment | added | domenico fiorenza | Hi Fosco (oh, I see I have revealed your secret name, something weird is going to happen to me like to Valerius Soranus, but anyway...), have you tried using the orthogonality between $D_{\geq 0}$ and $D_{\leq -1}$? | |
| Jun 4, 2014 at 19:00 | history | bounty ended | domenico fiorenza | ||
| Jun 4, 2014 at 10:56 | comment | added | fosco | I'm having some trouble in verifying that this defines a true factorization system on $\text{Ch}(R)$. In a few words it seems that your procedure defined a factorization functor $F\colon f\mapsto (f_L, f_R)$. But now do $\mathcal E = \{f\mid f_R\text{ iso}\},\mathcal M=\{f\mid f_L \text{ iso}\}$ form a factorization system? | |
| Jun 3, 2014 at 15:31 | comment | added | Eric Wofsey | Yeah, I didn't mean to imply that was any sort of harsh restriction. I don't know of any "natural" examples that don't satisfy that. | |
| Jun 3, 2014 at 15:29 | comment | added | domenico fiorenza | Perfect! a stable $\infty$-category with functorial truncation sequences and nullhomotopies was precisely the context I had in mind! Thanks a lot! | |
| Jun 3, 2014 at 10:36 | comment | added | Eric Wofsey | The functoriality of the nullhomotopy makes all the choices of maps in the second column canonical and functorial. | |
| Jun 3, 2014 at 10:36 | comment | added | Eric Wofsey | Actually, if I'm not mistaken, there is a subtlety that prevents my construction from working for an arbitrary triangulated category with t-structure. Namely, you cannot be sure that the composition $A\to C\to B$ is equal to the original map; all you know is that it is some map that commutes with the top and bottom cofiber sequences (but such maps are not unique). What you need is that everything lifts to an $\infty$-category in which the sequences $\tau_{\geq0}A\to A\to\tau_{\leq-1}A$ are functorial and there is also a functorial nullhomotopy of the composition. | |
| Jun 3, 2014 at 5:33 | vote | accept | domenico fiorenza | ||
| Jun 3, 2014 at 5:32 | comment | added | domenico fiorenza | Hi Eric, thanks! Indeed my candidate for $C$ was $C=\tau_{\leq -1}A\times_{\tau_{\leq-1}B}B$, which I see is the same $C$ you get, but I had been unable to show that $A\to C$ and $C\to B$ became equivalences under $\tau_{\leq -1}$ and $\tau_{\geq 0}$ by the fiber product description. Now I see how looking at the pullback diagram as a part of the big diagram gives that immediately by using $\tau_{\leq -1}\tau_{\geq 0}=0$ and $\tau_{\leq -1}\Sigma\tau_{\geq 0}=0$ and the 3-for-2 property for equivalences. And it also works for an arbitrary t-structure on a triangulated category, right? | |
| Jun 2, 2014 at 20:15 | history | answered | Eric Wofsey | CC BY-SA 3.0 |