The new exciting thing today is, the raising and lowering operator.

Let $E$ be the functor for adding strands, and $F$ the functor for removing strand, then we should

• define the functor $F$
• check that $F$ and $E$ are adjoint
• test it on T-branes.
• understand Rouquier's condition geometrically.

The setup is the following: $\gdef\lcal{\mathcal L}$

• we have a smooth closed surface $S$ with boundary, and there are some stops on the boundary.
• We can consider $k$ disjoint Lagrangians $L_1, \cdots, L_k$, which defines an object in $Sym^k(S) \RM \Delta$
• We can consider $k$ disjoint Lagrangians $\lcal_1, \cdots, \lcal_k$ in $S \times \C^*$, which defines an object in $Sym^k(S \times \C^*) \RM \Delta$. We assume these $\lcal_i = L_i \times F_i \In S \times \C^*$ are of product type.