  The purpose of this paper is to provide a unified method for dealing with
various 0-Hecke modules constructed using tableaux so far. To do this, we
assign a $0$-Hecke module to each left weak Bruhat interval, called a weak
Bruhat interval module. We prove that every indecomposable summand of the
$0$-Hecke modules categorifying dual immaculate quasisymmetric functions,
extended Schur functions, quasisymmetric Schur functions, and Young row-strict
quasisymmetric Schur functions is a weak Bruhat interval module. We further
study embedding into the regular representation, induction product,
restriction, and (anti-)involution twists of weak Bruhat interval modules.
