In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of B n with a representation of B n+1 . In this paper, we prove that this construction is functorial and can be extended: it inspires endofunctors, called Long-Moody functors, between the category of functors from Quillen's bracket construction associated with the braid groupoid to a module category. Then we study the effect of Long-Moody functors on strong polynomial functors: we prove that they increase by one the degree of very strong polynomiality.

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In this paper, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.

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.