====== 2024-06-29 ======

===== braiding functor / bimodule =====

If I have an $A-B$-bimodule $K$, and I have a right $A$-mod $M$, then I can do 
$$ M \otimes_A K$$
to get a right $B$-mod. And conversely, given a right $B$-mod $N$, I can do
$$ N \otimes_B K^\vee = Hom(K, N) $$
to get a right $A$-mod, assuming $K$ has enough dualizablity (whatever that means).

Consider the 1-strand KLRW algebra with $n$ punctures. In this case, the algebra before and after braiding are the same, but we should still call them $A_-$ (before) and $A_+$ (after). 

What is this bimodule? Webster would invent a new KLRW diagram with crossing red strands, and invent some rules.

How do you present a bimodule? I would say, as a chain complex of other bimodules.