Timeline for Complexes in stable categories
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 21, 2018 at 11:55 | vote | accept | CommunityBot | ||
| Dec 28, 2017 at 17:05 | comment | added | user13113 | @მამუკაჯიბლაძე: There is actually another diagram I've considered that the iterated cofiber construction works for, depicted below. Maybe this alternate diagram is better behaved; I think it can be seen as a subdiagram of the "upper triangular array" version of complex I mentioned in the OP. $$ \begin{matrix} A_n &\to& A_{n-1} &\to& A_{n-2} \\ & \nearrow \!\!\!\!\!\! \searrow & & \nearrow \!\!\!\!\!\! \searrow \\ 0 &\to& 0 &\to& 0 \end{matrix} $$ | |
| Dec 28, 2017 at 16:54 | comment | added | user13113 | @მამუკაჯიბლაძე: Yes; I'm assuming the new arrows all map to zero. Realization by way of iterated coequalizers doesn't actually require the new arrows to be zero; the point is to normalize a degree of freedom (since you can add the same map to both arrows of a parallel pair and get a new diagram satisfying the coequalizer condition and with the same realization). I'm not certain if it's enough to simply require each of the new arrows to be zero or if I need an additional constraint on the entire sequence of new arrows. | |
| Dec 28, 2017 at 8:07 | comment | added | მამუკა ჯიბლაძე | But how do you avoid conditions outside the index category? Don't you need to require that either some morphisms forcibly go to zero morphisms, or some object forcibly goes to the zero object? | |
| Dec 27, 2017 at 21:35 | comment | added | user13113 | @მამუკაჯიბლაძე: I asked my question because I hadn't appreciated the difference either! I don't yet have a good intuition yet, but mechanically the difference is that it's an equation in the index category, so a functor introduce more equivalences than I had originally. That the iterative calculation works out follows from this procedure involving Kan extensions (expressed there for limits rather than colimits). Or I could be making errors someplace. | |
| Dec 27, 2017 at 16:48 | comment | added | მამუკა ჯიბლაძე | I don't understand the last part. Some $d$ coequalizes $(d',0)$ if and only if $dd'=0$, so how does this differ from your original version? | |
| Dec 27, 2017 at 15:14 | history | edited | user13113 | CC BY-SA 3.0 |
added 74 characters in body
|
| Dec 27, 2017 at 14:00 | history | edited | user13113 | CC BY-SA 3.0 |
added 228 characters in body
|
| Dec 26, 2017 at 23:14 | history | answered | user13113 | CC BY-SA 3.0 |