In this paper, we generalize the principle of Long-Moody construction for representations braid groups to other groups, such as mapping class surfaces. Namely, introduce endofunctors over a functor category that encodes family groups. They are called functors and provide new representations. context, notions polynomial defined play an important role in study homological stability. We prove that...

We show that a derivator is stable if and only homotopy finite limits colimits commute, limit functors have right adjoints, colimit left adjoints. These characterizations generalize to an abstract notion of "stability relative class functors", which includes in particular pointedness, semiadditivity, ordinary stability. To prove them, we develop the theory derivators enriched over monoidal weig...