We construct local coordinates for the Weinstein groupoid of a non-integrable Lie algebroid. To this end, we reformulate the notion of bi-submersion in a completely algebraic way and prove the existence of bi-submersions as such for the Weinstein groupoid. This implies that a C*-algebra can be attached to every Lie algebroid.

We establish an integration theory for singular subalgebroids, by diffeological groupoids. To do so, we single out a class of diffeological groupoids satisfying specific properties, and we introduce a differentiation-integration procedure under which they correspond to singular subalgebroids. Our definition of integration distinguishes the holonomy groupoid from the graph, although both differentiate to the original singular subalgebroid: the holonomy groupoid satisfies a certain submersive property, while the graph does not.