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