Timeline for Connections in terms of tangent ($\infty$-)categories?
Current License: CC BY-SA 3.0
11 events
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| Mar 18, 2018 at 12:39 | history | edited | user20948 | CC BY-SA 3.0 |
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| Mar 18, 2018 at 9:19 | comment | added | user20948 | @JasonStarr Pretty nice! That explains algebraically why Chern classes don't depend on the choice of connection: naively speaking, the boundary map associated to a s.e.s. doesn't depend on the choice of preimages. Previously I only knew that the first Chern class could be described by the s.e.s $\mathbb C\to\mathscr O_X\to\mathscr O_X^*$. Pretty interesting! | |
| Mar 18, 2018 at 9:06 | comment | added | user20948 | @JasonStarr Maybe you are referring to page 244, where he said that «On vérifie en effet facilement que les splittages sont les connexions sur $M$ relativement à $S$». Thank you very much. Since I don't know Atiyah class, etc, I need more time to understand them. Now seemingly this is just a silly question. | |
| Mar 18, 2018 at 8:38 | comment | added | user20948 | @JasonStarr Thanks. Are you referring to (1.2.6.3) by Atiyah exact sequence, or stacks.math.columbia.edu/tag/09CH? | |
| Mar 18, 2018 at 8:33 | history | edited | user20948 | CC BY-SA 3.0 |
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| Mar 16, 2018 at 19:41 | comment | added | mme | @TimCampion In differential geometry, a useful fact is that holonomy is a equivariant under gauge transformations. Perhaps a similar equivariance plays a role here. | |
| Mar 16, 2018 at 17:37 | comment | added | Tim Campion | Of course, it's also true in differential geometry that the space of connections on a vector bundle is contractible (in fact, it's an affine space), but that's okay because differential geometry is not homotopy-invariant, so it makes sense for the connection to store information which is not homotopy-invariant. On the other hand, infinity categories are "homotopy-invariant" in the sense that any meaningful notion is invariant under equivalence of infinity categories. | |
| Mar 16, 2018 at 17:32 | comment | added | Tim Campion | The thing I find confusing about this is that I like to think of a connection as a path-lifting operation. From this perspective, any (co)cartesian fibration comes equipped with a unique "connection" given by taking (co)cartesian lifts. This is analogous to the fact that any Serre fibration has a path-lifting operation which is unique up to coherent homotopy. A connection in terms of a Leibniz rule is supposed to just be an "infinitesimal path-lifting operation", but if the space of connections is contractible, this leaves me puzzled as to what information could be stored in the connection. | |
| Mar 16, 2018 at 14:01 | history | edited | user20948 | CC BY-SA 3.0 |
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| Mar 16, 2018 at 11:26 | comment | added | Jason Starr | Isn't a connection a splitting of the Atiyah exact sequence? If so, this is discussed at length near the beginning of volume I of Illusie's "Complexe Cotangent et Deformations" where he discusses properties of the functor $P$. | |
| Mar 16, 2018 at 9:56 | history | asked | user20948 | CC BY-SA 3.0 |