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higher connections; higher parallel transport; generalised Atiyah groupoid; generalised Atiyah sequence; orthogonal factorisation systems
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.
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[2] Berwick-Evans, D., de Brito, P.B., Pavlov, D.: Classifying spaces of infinity-sheaves. 2019. DOI: DOI 10.48550/ARXIV.1912.10544
[3] Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories. Handbook of algebraic topology 73 (1995), 126. MR 1361887
[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: DOI 10.1142/S0129055X16500124 | MR 3535115
[5] Friedman, G.: Survey article: an elementary illustrated introduction to simplicial sets. Rocky Mountain J. Math. (2012), 353–423. MR 2915498
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