The pro-Chern-Schwarz-MacPherson class in Borel-Moore motivic homology

We show that the zero-dimensional part of the pro-Chern-Schwarz-MacPherson class defined by Aluffi is equal to the pro-characteristic class in limit Borel-Moore motivic homology. A similar construction also produces a quadratic refinement of this class in the limit Borel-Moore Milnor-Witt homology. This is a joint work with Peng Sun and Enlin Yang.