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We define a type B analogue of the category of finite sets with surjections,and we study the representation theory of this category. We show that theopposite category is quasi-Grobner, which implies that submodules of finitelygenerated modules are again finitely generated. We prove that the generatingfunctions of finitely generated modules have certain prescribed poles, and weobtain restrictions on the representations of type B Coxeter groups that canappear in such modules. Our main example is a module that categorifies thedegree i Kazhdan-Lusztig coefficients of type B Coxeter arrangements.

A type B analogue of the category of finite sets with surjections