the category {\bf fi} and its variants have been of great interest recently. being a finitely generated {\bf fi}-module implies many desirable properties about sequences of symmetric group representations, in particular representation stability. we define a new combinatorial category analogous to {\bf fi} for the 0-hecke algebra, denoted by $\mathcal{h}$, indexing sequences of representations of $h_n(0)$ as $n$ varies under suitable compatibility conditions. we then provide examples of $\mathcal{h}$-modules and use these to discuss some properties finitely generated $\mathcal{h}$-modules possess, including a new form of representation stability and eventually polynomial growth.