Weak Bruhat interval modules of finite-type $0$-Hecke algebras and projective covers

We extend the recently-introduced weak Bruhat interval modules of the type A $0$-Hecke algebra to all finite Coxeter types. We determine, in a type-independent manner, structural properties for certain general families of these modules, with a primary focus on projective covers and injective hulls. We apply this approach to recover a number of results on type A $0$-Hecke modules in a uniform way, and obtain some additional results on recently-introduced families of type A $0$-Hecke modules.