The braid variety of a positive braid and the augmentation variety of a Legendrian link both admit decompositions coming from weaves and rulings, respectively. In joint work with Johan Asplund, James Hughes, Caitlin Leverson, Wenyuan Li and Angela Wu, we prove that these decompositions agree under an isomorphism between the braid variety and the augmentation variety. We also prove that both decompositions coincide with a Deodhar decomposition and another decomposition coming from the microlocal theory of sheaves. Our proof relies on a detailed comparison between weaves and Morse complex sequences.