Integration of (higher) algebroids, (higher) principal bundles with connection and (higher) differential characters

I will discuss the correspondence between differential characters and higher principal $S^1$-bundles with connection, via the integration of Lie algebroids. The main point of the talk is that one needs to consider an "homological" integration, different from the standard "homotopic" integration. This is true already for ordinary principal $S^1$-bundles where the relevant integration is the so called genus integration of the prequantum algebroid, introduced in joint work with Contreras.