| Title:
|
Generalised Atiyah’s theory of principal connections (English) |
| Author:
|
Nárožný, Jiří |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2022 |
| Pages:
|
241-256 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This is a condensed report from the ongoing project aimed on higher principal connections and their relation with higher differential cohomology theories and generalised short exact sequences of $L_\infty $ algebroids. A historical stem for our project is a paper from sir M. Atiyah who observed a bijective correspondence between data for a horizontal distribution on a fibre bundle and a set of sections for a certain splitting short exact sequence of Lie algebroids, nowadays called the Atiyah sequence. In a meantime there was developed quite firm understanding of the category theory and in the last two decades also the higher category/topos theory. This conceptual framework allows us to examine principal connections and higher principal connections in a prism of differential cohomology theories. In this text we cover mostly the motivational part of the project which resides in searching for a common language of these two successful approaches to connections. From the reasons of conciseness and compactness we have not included computations and several lengthy proofs. (English) |
| Keyword:
|
higher connections |
| Keyword:
|
higher parallel transport |
| Keyword:
|
generalised Atiyah groupoid |
| Keyword:
|
generalised Atiyah sequence |
| Keyword:
|
orthogonal factorisation systems |
| MSC:
|
18N60 |
| idZBL:
|
Zbl 07655746 |
| idMR:
|
MR4529816 |
| DOI:
|
10.5817/AM2022-4-241 |
| . |
| Date available:
|
2022-11-28T12:25:07Z |
| Last updated:
|
2023-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151150 |
| . |
| Reference:
|
[1] Atiyah, M.F.: Complex analytic connections in fibre bundles.Trans. Amer. Math. Soc. 85 (1957), no. 1, 181–207. MR 0086359, 10.1090/S0002-9947-1957-0086359-5 |
| Reference:
|
[2] Berwick-Evans, D., de Brito, P.B., Pavlov, D.: Classifying spaces of infinity-sheaves.2019. DOI: http://dx.doi.org/10.48550/ARXIV.1912.10544 10.48550/ARXIV.1912.10544 |
| Reference:
|
[3] Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories.Handbook of algebraic topology 73 (1995), 126. MR 1361887 |
| Reference:
|
[4] Fiorenza, D., Rogers, C.L., Schreiber, U.: Higher U(1)-gerbe connections in geometric prequantization.Rev. Math. Phys. 28 (2016), no. 06, 1650012. DOI: http://dx.doi.org/10.1142/s0129055x16500124 MR 3535115, 10.1142/S0129055X16500124 |
| Reference:
|
[5] Friedman, G.: Survey article: an elementary illustrated introduction to simplicial sets.Rocky Mountain J. Math. (2012), 353–423. MR 2915498 |
| . |
