Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with effective motives. My questions are:
Does there exist any 'philosophical' concept of effectivity related with these examples?
Do there exist any other parallel notions of effectivity (for Galois representations? for Hodge modules? for mixed motivic sheaves?)?