  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 the
opposite category is quasi-Grobner, which implies that submodules of finitely
generated modules are again finitely generated. We prove that the generating
functions of finitely generated modules have certain prescribed poles, and we
obtain restrictions on the representations of type B Coxeter groups that can
appear in such modules. Our main example is a module that categorifies the
degree i Kazhdan-Lusztig coefficients of type B Coxeter arrangements.
