  We construct a finite dimensional quiver algebra from the non-simply laced
type $B$ Dynkin diagram, which we call the type $B$ zigzag algebra. This leads
to a faithful categorical action of the type $B$ braid group $\mathcal{A}(B)$,
acting on the homotopy category of its projective modules. This categorical
action is also closely related to the topological action of $\mathcal{A}(B)$,
viewed as mapping class group of the punctured disc -- hence our exposition can
be seen as a type $B$ analogue of Khovanov-Seidel's work in
arXiv:math/0006056v2. Moreover, we show that certain category of bimodules over
our type $B$ zigzag algebra is a quotient category of Soergel bimodules,
resulting in an alternative proof to Rouquier's conjecture on the faithfulness
of the 2-braid groups for type $B$.
