Categorifications of rational Hilbert series and characters of FSop modules

We introduce a method for associating chain complex to module over combinatorial category, such that if the is exact then has rational Hilbert series. prove homology--vanishing theorems these complexes several categories including: category of finite sets and injections, opposite surjections, dimensional vector spaces field injections. Our main applications are modules known as $FS^{op}$ modules. obtain many constraints on sequence symmetric group representations underlying finitely generated module. In particular, we describe its character in terms functions call exponentials. results have new consequences homology moduli space stable marked curves, equivariant Kazhdan-Luzstig polynomial braid matroid.