Recently, based on a new procedure to quantize the theory in the continuum, it was argued that Singer's theorem points toward the existence of a Yang-Mills ensemble. In the new approach, the gauge fields are mapped into an auxiliary field space used to initially determine sectors labeled by center vortices, and then separately fix the gauge on them. In this work, we study this procedure in more detail. We provide examples of configurations belonging to these sectors and discuss the existence of non-Abelian degrees of freedom. Then, we discuss the importance of the mapping injectivity, and show that this property holds infinitesimally for typical configurations of the vortex-free sector and for the simplest example in the one-vortex sector. Finally, we show that these examples are free from Gribov copies.