Generalised Atiyah’s theory of principal connections

This is a condensed report from the ongoing project aimed on higher principal connections and their relation with differential cohomology theories generalised short exact sequences of $L_\infty $ algebroids. A historical stem for our paper sir M. Atiyah who observed bijective correspondence between data horizontal distribution fibre bundle set sections certain splitting sequence Lie algebroids, nowadays called sequence. In meantime there was developed quite firm understanding category theory in last two decades also category/topos theory. conceptual framework allows us to examine prism theories. this text we cover mostly motivational part which resides searching common language these successful approaches connections. From reasons conciseness compactness have not included computations several lengthy proofs.