In this talk I will provide a brief report from the ongoing project aimed on higher principal connections and their relation with higher differential cohomology theories and generalised short exact sequences of higher Lie 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.
The goal of my talk is to present this differential cohomology framework for defining principal connections and higher principal connections, and to show some general mechanisms for reformulating these data into a form that mimics Atiyah's theory.