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