Timeline for Putting sheaves to work for algebraic topology?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8, 2021 at 23:06 | vote | accept | Siddharth Bhat | ||
| Mar 8, 2021 at 16:08 | comment | added | Sam Gunningham | @DmitiriPavlov I guess there are really 2 independent flavours of correspondence: firstly, between fibrations and representations (parallel transport construction) and secondly between infinitesimal and "honest" parallel transport (integration). The infinitesimal/honest distinction can be quite large in algebraic and analytic geometry. e.g. D-modules have infinitesimal parallel transport by definition, but only very special ones (holonomic) admit parallel transport along "honest" paths. I realize that you are coming more from homotopy theory though, so these comments are kind of off-topic | |
| Mar 8, 2021 at 16:06 | comment | added | Sam Gunningham | @DmitiriPavlov Yes I agree with what you are saying (for a suitable notion of germ). My comment is really just about what aspect of this situation the term "Riemann-Hilbert correspondence" refers to. Perhaps the terminology is a culture dependent to some extent. | |
| Mar 8, 2021 at 15:58 | comment | added | Denis Nardin | @DmitriPavlov I said "right" because some people ask the decomposition $p^{-1}U=\coprod_\alpha U_\alpha$ to be the path-connected component decomposition (which breaks the whole theory utterly in the case of not locally path-connected spaces). It is clear that the issue is another one here (I still think the definition I gave in my comment above is the "standard" one though, "right" is ofc a matter of preference). | |
| Mar 8, 2021 at 15:56 | comment | added | Dmitri Pavlov | @DenisNardin: A potential point of confusion may have arisen when you referred to the "right definition of covering spaces as fiber bundles with discrete fibers". I assumed you meant to use a nonequivalent (i.e., "right") definition, but perhaps it is best to clarify what exactly you mean here. | |
| Mar 8, 2021 at 15:23 | comment | added | Dmitri Pavlov | @SamGunningham: But you do have (the appropriate analogue of) infinitesimal parallel transports: for a covering T→B, a map F→T with an extension of F→T→B to a germ F' of F admits a unique extension to a map F'→T. Germs take over finite-order or formal neighborhoods in this setting, so this is the appropriate analogue of infinitesimal parallel transport. | |
| Mar 8, 2021 at 15:20 | comment | added | Dmitri Pavlov | @DenisNardin: The definition I used above is that a fiber bundle with a discrete fiber F should be a morphism of stacks X→B(Aut(F)), which implies the condition I pointed above. Of course, different definitions are possible. | |
| Mar 8, 2021 at 15:18 | comment | added | Sam Gunningham | For example, the term Riemann-Hilbert correspondence is used to describe the equivalence between the moduli space of flat holomorphic connections and the representations of the fundamental group of a compact Riemann surface. The latter I would freely identify with locally constant sheaves or fiber bundles with discrete fibers. But the former is of a different nature: the moduli space carries a different algebraic structure, and the equivalence is strictly transcendental in nature. | |
| Mar 8, 2021 at 15:13 | comment | added | Sam Gunningham | @DmitiriPavlov: Sure, under some conditions all these things are equivalent. But the defining feature of the Riemann-Hilbert correspondence to my mind is the passage between the infinitesimal notion of parallel transport (i.e. a covariant derivative) and the actual positive-time parallel transport (i.e. a representation of the fundamental group). This is just a matter of terminology though. :) | |
| Mar 8, 2021 at 15:12 | comment | added | Denis Nardin | Hrmm... maybe I'm using a different notion of "fiber bundle". Isn't a covering space just literally a space $p:Y\to X$ such that for every point $x\in X$ there's a nbd $U$ and a homeomorphism over $U$ $f^{-1}U\cong U\times F_x$ for some discrete space $F_x$? Or are we working with a different definition? (incidentally my definition of fiber bundle is the same, except that $F$ is not required to be discrete) | |
| Mar 8, 2021 at 15:10 | comment | added | Dmitri Pavlov | @DenisNardin: The missing part is the proof that (after refining the cover) the transition maps can be made constant, which is required for a fiber bundle with a discrete fiber. | |
| Mar 8, 2021 at 15:08 | comment | added | Denis Nardin | @DmitriPavlov I'm sorry but I am confused - why this notion of locally constant sheaf does not correspond to covering spaces? There's an equivalence between sheaves over $X$ and local homeomorphisms $Y→X$ given by the étalé space construction, and I thought that locally constant sheaves correspond exactly to covering spaces under this equivalence. I cannot find a discussion of this at the linked nCafé page. What am I missing? | |
| Mar 8, 2021 at 15:05 | comment | added | Dmitri Pavlov | @SamGunningham: Covering spaces resemble flat connections: both admit a notion of a parallel transport that is invariant under homotopies of paths. Both have a total space that projects to the base space. | |
| Mar 8, 2021 at 15:04 | comment | added | Dmitri Pavlov | @DenisNardin: What other definition of a covering space do you have in mind? From my point of view, the problem is that there are two nonequivalent definitions of a locally constant sheaf, and the one commonly given in the literature (a sheaf that is locally isomorphic to the constant sheaf) simply does not give a category equivalent to covering spaces, as explained in the linked nCafé post. | |
| Mar 8, 2021 at 15:04 | comment | added | Sam Gunningham | This is a minor comment, but to my mind all three of those categories live on the same side of the Riemann-Hilbert correspondence. The other side would consist of some form of flat connections (which only really makes sense in a differential/algebro-geometric context with linear coefficients). I guess I would call the equivalence of the first two categories the Galois correspondence or something. | |
| Mar 8, 2021 at 9:23 | comment | added | Konrad Waldorf | A complete discussion of the equivalence between the first two categories (in german) is in the topology textbook by Laures and Szymik. | |
| Mar 8, 2021 at 7:32 | comment | added | Denis Nardin | Aren't the last two categories always equivalent (at least if you take the right definition of covering spaces as fiber bundles with discrete fibers)? | |
| Mar 7, 2021 at 23:21 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 35 characters in body
|
| Mar 7, 2021 at 23:16 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |