Timeline for What is a higher derived constructible sheaf
Current License: CC BY-SA 3.0
11 events
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| Apr 24, 2013 at 18:25 | comment | added | Dmitry Vaintrob | @David So it sounds like you're saying (in the algebraic case), that the 'etale topos contains all the topology of a manifold up to some sort of profinite completion, and higher-category analogues of locally constant (resp. constructible) sheaves are well-approximated by sheaves on this topos on the one hand, and therefore by D-modules on the other hand. Is this correct? | |
| Apr 24, 2013 at 3:59 | comment | added | Sam Gunningham | @David: I agree about the finiteness issues. I only mention this, as in the question, the $C_\ast(\Omega X)$-version of local systems was talked about. And it happens to be an issue that I am interested in myself! | |
| Apr 24, 2013 at 2:10 | comment | added | David Ben-Zvi | Thanks Sam! (you keeping tell me this fact enough times I might start to remember it!) OTOH unless you're imposing some such finiteness conditions on your sheaves it's probably a bad idea to call them constructible.. In any case the issue is size (as you say already in the case of the circle), not really anything to do with particularly "infinity" issues. | |
| Apr 24, 2013 at 1:41 | comment | added | Sam Gunningham | Even in the simply connected case, the two versions of $\infty$-local systems are not quite the same. As I commented below, they correspond to $C_\ast(\Omega X)$ vs $C^\ast(X)$ modules. These categories are closely related (by some version of Koszul duality), but not quite the same. | |
| Apr 24, 2013 at 1:31 | comment | added | Sam Gunningham | @David: I think you need to be careful about finiteness issues. For example, local systems on $\mathbb C^\times$ in Dmitry's sense would be representations of $\pi_1 (\mathbb C^\times)$. But you won't be able to see the indecomposible infinite dimensional representations of $\mathbb Z$ using $D$-modules (I think the ``de Rham homotopy type'' should only be able to the pro-algebraic completion of $\pi_1$, or something along those lines...) | |
| Apr 24, 2013 at 0:43 | comment | added | David Ben-Zvi | I think you need X connected for your description of local systems, but the coefficients are arbitrary (i.e., $k$ could be the sphere). But the same category can also be said concretely as complexes of sheaves with locally constant cohomology groups. Then the $\infty$-version of constructible sheaves is just the $\infty$-category of complexes of sheaves with constructible cohomology. And indeed that's derived equivalent to the $\infty$-category of regular holonomic $\D$-modules in the complex analytic setting. | |
| Apr 23, 2013 at 23:54 | vote | accept | Dmitry Vaintrob | ||
| Apr 23, 2013 at 23:49 | vote | accept | Dmitry Vaintrob | ||
| Apr 23, 2013 at 23:50 | |||||
| Apr 23, 2013 at 22:58 | answer | added | Sam Gunningham | timeline score: 6 | |
| Apr 23, 2013 at 21:23 | history | edited | Dmitry Vaintrob | CC BY-SA 3.0 |
minor edits + added local-systems tag
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| Apr 23, 2013 at 20:55 | history | asked | Dmitry Vaintrob | CC BY-SA 3.0 |