I'll begin with some dreams and motivation regarding lifting notions of lattice theory to homological algebra. For most of the talk, we'll study a concrete example: the lattices of integer cuts and flows associated to a finite graph. Given a graph G and choice of spanning tree T, we construct an algebra A(G,T) such that the Groethendieck group of the category of finitely generated A(G,T)-modules with the Euler (Ext) pairing contains the cut and flow lattices of G as orthogonal complements. We'll discus many open problems regarding generalisations and possible uses for the extra structure that is present at the category level, such as gradings. Joint work in progress with Anthony Licata.