Classifying Spaces of Diffeological Groups

Irrational tori are groups that show up as, for example, quotients of the real numbers modulo a proper subgroup. If the orbits are dense, this quotient has trivial topology. While seemingly pathological, this group shows up quite often in mathematics: for instance, the quotient of the torus by the irrational Kronecker flow, or in geometric quantisation and the integration of certain Lie algebroids as the structure group of certain principal bundles. Can we give the space any type of useful "smooth" group structure so that we can treat it similarly as a Lie group? The answer is "yes": its natural diffeological group structure. With this structure, we can talk about its classifying space, and construct connection 1-forms on principal bundles. In this talk we shall do so for a diffeological group G, and mention a few properties one would hope for given the well-established theory for topological groups.