In Remarks 2.4.3 and 2.4.4 we have observed that a prebraiding need not satisfy the braid relation, and that the braid relation in prebraidings on identity functors can be verified in two ways using naturality. On the level of the homotopy category of Soergel bimodules, this is reflected in the fact, that the chain maps implementing the homotopy equivalence corresponding to a braid relation live in a 2-dimensional space [EK10, Section 3, Reidemeister 3 generators]. One dimension is encoding an overall non-zero scaling. Even after fixing this, there exists however a 1-dimensional affine subspace of representatives of the same homotopy class of chain maps. Two distinct (and thus spanning) points in this subspace can be built from slide chain maps, analogously to the two ways of establishing the braid relation in Remark 2.4.4.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2