Geodesic
Generalised Atiyah’s theory of principal connections
Archivum mathematicum, Tome 58 (2022) no. 4, pp. 241-256

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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.
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.
DOI : 10.5817/AM2022-4-241
Classification : 18N60
Keywords: higher connections; higher parallel transport; generalised Atiyah groupoid; generalised Atiyah sequence; orthogonal factorisation systems 
Nárožný, Jiří. Generalised Atiyah’s theory of principal connections. Archivum mathematicum, Tome 58 (2022) no. 4, pp. 241-256. doi: 10.5817/AM2022-4-241
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