About dual cube theorems

Most of the properties of the category of Lusternik–Schnirelmann come from the cube theorems of Mather, especially the second. The duals of these theorems are false, which makes the dual category problematic. However, a weaker version of the second cube theorem is sufficient to get the usual properties of the LS-category. In this note, we show that this weaker version of the dual of the first theorem is false. The weaker version of the dual of the second cube theorem remains an open problem.