Address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18000 Praha 8, Czech Republic
E-mail: narozny@karlin.mff.cuni.cz
Abstract: 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.
AMSclassification: primary 18N60.
Keywords: higher connections, higher parallel transport, generalised Atiyah groupoid, generalised Atiyah sequence, orthogonal factorisation systems.
DOI: 10.5817/AM2022-4-241