Generalized Long–Moody functors

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, under additional assumptions, increases very strong weak degrees by one.