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

Curves in the disc, the type B braid group, and the type B zigzag algebra