Timeline for Are simplicial abelian sheaves fibrant?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 3, 2020 at 13:59 | comment | added | Dmitri Pavlov | @NanjunYang: The Brown–Gersten model structure is the projective model structure on simplicial sheaves. Everything I said about the projective and local projective model structures applies also to the Brown–Gersten model structure. | |
| Mar 3, 2020 at 9:15 | comment | added | Nanjun Yang | My final decision is to use the Brown-Gersten model structure. I think $K(A,n)$ is fibrant if $A$ is injective. | |
| Feb 28, 2020 at 15:46 | comment | added | Dmitri Pavlov | @NanjunYang: In the motivic context, you would need K(A,n)=Γ(A[n]) to be locally injectively fibrant and not just injectively fibrant. The argument in my post shows that K(A,n)=Γ(A[n]) is not a local object, so cannot be locally injectively fibrant. Even ignoring locality, though, I am pretty sure that K(A,n)=Γ(A[n]) is not injectively fibrant. Why not use the projective local model structure instead, where the Eilenberg–MacLane spectra seem to be fibrant? | |
| Feb 28, 2020 at 5:38 | comment | added | Nanjun Yang | Maybe $K(A,n)$ isn't injectively fibrant in general. I think I could avoid this statement. | |
| Feb 28, 2020 at 4:30 | comment | added | Nanjun Yang | So are the simplicial abelian sheaves still fibrant under injective model structure? I want to prove that the Eilenberg MacLane spectra are stably fibrant finally, thus I have to show that $K(A,n)$ is injectively fibrant. I've been using injective model structures. | |
| Feb 28, 2020 at 4:26 | vote | accept | Nanjun Yang | ||
| Feb 28, 2020 at 4:07 | comment | added | Dmitri Pavlov | @NanjunYang: The generators you wrote down are precisely the generating cofibrations for the projective and local projective model structures. The prevailing opinion with respect to the generating cofibrations for the injective model structure is that there is no explicit such a set of generators, excluding some special categories of presheaves (but Sm/k is not one of them). | |
| Feb 27, 2020 at 23:19 | comment | added | Nanjun Yang | I mean the injective model structure. That is why I want to find a system of generators. | |
| Feb 27, 2020 at 19:58 | comment | added | Dmitri Pavlov | @SimonHenry: I added a specific example. | |
| Feb 27, 2020 at 19:57 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 558 characters in body
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| Feb 27, 2020 at 19:29 | comment | added | Simon Henry | I definitely agree with the first line. but I don't quite see the relation between being fibrant in the local projective model structure and having trivial sheaf cohomology. | |
| Feb 27, 2020 at 19:00 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |