Representation stability for sequences of 0-Hecke modules

We define a new category analogous to ${\bf FI}$ for the $0$-Hecke algebra $H_n(0)$ called category, $\mathcal{H}$, indexing sequences of representations as $n$ varies under suitable compatibility conditions. establish type representation stability in this setting and prove it is implied by being finitely generated $\mathcal{H}$-module. then provide examples $\mathcal{H}$-modules discuss further desirable properties these modules possess.