Abstrakt
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. 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/top os 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.