Timeline for Model structure on spaces with local coefficients
Current License: CC BY-SA 3.0
11 events
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| Oct 9, 2016 at 13:25 | comment | added | Karol Szumiło | @local The simplicial set I associated with $(X, A)$ is not actually a simplicial abelian group and it does not detect local homology isomorphisms (as far as I can tell). You would have to suitably close it up under linear combinations, but then the resulting functor is no longer a right adjoint and it's not clear how to use it to create a model structure. I still think that existence of such structure is plausible, but probably it would have to be constructed using methods that Bousfield originally used to construct non-local homology localizations. | |
| Oct 9, 2016 at 11:30 | comment | added | local | @KarolSzumiło I read your answer. What was wrong about it? | |
| Oct 8, 2016 at 10:08 | comment | added | Karol Szumiło | I did add an answer, but deleted it later since I realized it was incorrect and didn't know how to fix it. | |
| Oct 7, 2016 at 22:08 | comment | added | Karol Szumiło | @DenisNardin You are right. I got a little confused by these pushforwards at first, but they are just left Kan extensions along the induced functors between fundamental groupoids. In that case the construction of the model structure seems rather easy. I will add an answer. | |
| Oct 7, 2016 at 20:44 | comment | added | Denis Nardin | @KarolSzumiło It is the total category of the Grothendieck fibration classified by the functor sending $X$ to the category of local systems on $X$, so it is cocomplete by general principles (in particular the pushouts are pushouts in spaces together with the pushout of the pushforwards of the local systems). | |
| Oct 7, 2016 at 20:02 | comment | added | Karol Szumiło | Is this category cocomplete? It's fairly easy to construct filtered colimits, but I don't see why pushouts exist. | |
| Oct 7, 2016 at 18:54 | comment | added | local | No. Just a natural transformation, not necessarily an isomorphism. I edited the question accordingly. | |
| Oct 7, 2016 at 18:53 | history | edited | local | CC BY-SA 3.0 |
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| Oct 7, 2016 at 16:47 | comment | added | Denis Nardin | Can you clarify what are the maps in this category? Precisely, if $f:(X,A)→(Y,B)$ is a map, are you requiring that $A→f^*B$ is an isomorphism of local systems? | |
| Oct 7, 2016 at 16:33 | review | First posts | |||
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| Oct 7, 2016 at 16:31 | history | asked | local | CC BY-SA 3.0 |