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**Generalised Atiyah’s theory of principal connections.**
*(English)*
Zbl 07655746

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. F. Atiyah [Trans. Am. Math. Soc. 85, 181–207 (1957; Zbl 0078.16002)] 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.

### MSC:

18N60 | \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories |

### Keywords:

higher connections; higher parallel transport; generalised Atiyah groupoid; generalised Atiyah sequence; orthogonal factorisation systems### Citations:

Zbl 0078.16002
Full Text:
DOI

### References:

[1] | Atiyah, M. F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), no. 1, 181-207 · Zbl 0078.16002 · doi:10.1090/S0002-9947-1957-0086359-5 |

[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 · doi:10.48550/ARXIV.1912.10544 |

[3] | Dwyer, W. G.; Spalinski, J., Homotopy theories and model categories, Handbook of algebraic topology 73 (1995), 126 · Zbl 0869.55018 |

[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 · Zbl 1366.53068 · doi:10.1142/S0129055X16500124 |

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