Timeline for Extraordinary cohomology as a derived functor?
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| Jul 20, 2010 at 0:32 | comment | added | Anatoly Preygel | The proof of the Theorem demonstrates how to get somewhat more "hands on" with sheaves of spaces (esp. 7.1.3). Another thing to keep in mind: The first step from sheaves of sets to sheaves of spaces, is sheaves of groupoids. This demonstrates the choices one can make in defining "presheaves" (fibered categories vs not), and what the higher bits of the sheaf condition encode. I don't know of a specific reference for sheaves of spectra, but the definitions are analogous to the space versions. (Also, I should repeat that the $R$ in $Rf_*$ and $R\Gamma$ was just analogy; it'd be $f_*$ in HTT.) | |
| Jul 20, 2010 at 0:16 | comment | added | Anatoly Preygel | I don't know a to-the-point reference, and the definitions are as in the classical case once the background is laid. For sheaves of spaces (for a Grothendieck topology, like say a usual topology) a definition appears in HTT 6.2.2. A better place to start looking may be HTT Section 7.1. Theorem 7.1.0.1 encodes, upon unraveling it, the relationship between the sheafy and representable perspectives (for sheaves of spaces). (Pullback from a point gets you the "constant sheaf", and pushforward to a point is what I was calling $R\Gamma$. The $R$ was just by analogy, though.) .. (cont.) .. | |
| Jul 19, 2010 at 20:00 | comment | added | algori | Anatoly -- thanks, but could you please give more details or a more specific reference? I'm sure there is plenty of good stuff to be found in Lurie's Higher topos theory or in Appendix A to DAG VI but it is still not clear to me what exactly the category is that $E$ is an object of. Moreover, how one defines $Rf_{\ast}$ and $R\Gamma$ for sheaves of spectra? So far the beginning of your argument looks exactly like what one would do in the classical case, with abelian groups replaced with spectra. | |
| Jul 17, 2010 at 3:49 | comment | added | Anatoly Preygel | Finally, I should mention that in this example we only ever needed to work with "locally constant" sheaves of spectra. Just as locally constant sheaves of e.g., vector spaces admit a simpler description on nice spaces (as "local systems"), so it is for spectra. Details are in above refs, but here's the intuitive sketch: At each point, you have a spectrum. For each path, you have an equivalence between the spectra over them; for any two composable paths, you have a way of "filling in the triangle" (i.e., a homotopy between the two resulting maps); etc. | |
| Jul 17, 2010 at 3:43 | comment | added | Anatoly Preygel | .. some combination of Lurie's "Higher Topos Theory" (with more good stuff in Appendix A to DAG VI), or for a more classical perspective something like May-Sigurdsson "Parameterized homotopy theory." What I call "sheaves of spectra" would go under "ex-spectra" in those terms. | |
| Jul 17, 2010 at 3:41 | comment | added | Anatoly Preygel | The "short answer" is you start with a(n infinity-)category of presheaves of spectra, and impose descent with respect to covers. (What's a sheaf? For any cover, the map from $\mathcal{F}(U)$ to a certain equalizer is an isomorphism. The infinity-cat. version will be that the map from $\mathcal{F}(U)$ to the holim of a Cech-type cosimplicial diagram is an equivalence.) If enough infinity-categorical machinery is in place, this is as much of a definition as the usual definition of a sheaf (of sets/abelian groups). You can find as much detail as you want in ... (continued) ... | |
| Jul 17, 2010 at 3:20 | comment | added | algori | Thanks, Anatoly! This sounds very interesting but what exactly is the category of sheaves of spectra? | |
| Jul 17, 2010 at 2:46 | history | answered | Anatoly Preygel | CC BY-SA 2.5 |