The paper introduces a general model for the study of decomposability in abstract argumentation, i.e. the possibility of determining the semantics outcome based on local evaluations in subframeworks. As such, the paper extends a previous work by generalizing over the kind of information locally exploited. While not concerned with specific semantics, the paper shows the range of decomposable semantics with varying degrees of local information. It also introduces the notion of a canonical local function, which can enforce decomposability whenever this is possible.