The Long-Moody construction and polynomial functors

Generalized Long-Moody functors

“…Then, we define the generalized Long-Moody functors and establish some of their first properties in Section 2.2. In addition of recovering all the results of [30, Section 2] (see Section 3.5), we give a new approach to the tools and conditions previously considered in [30], allowing a wider application and a deeper understanding of these constructions.…”

“…A braided monoidal category is automatically pre-braided. However, a pre-braided monoidal category is not necessarily braided (see for example [30,Remark 1.15]).…”

“…In this case, the Artin representations {a n,1 : B n → Aut (F n )} n∈N , defined by the action B n on the fundamental group π 1 Σ n 0,1 ∼ = F n for all natural numbers n, assemble to define a functor A β 1 : Uβ → Gr (see [30,Example 2.3] or Section 3.2). Moreover, there exists a family of non-trivial morphisms {ς n,1 : [30,Example 2.1] or Definition 3.17) such that the morphism given by the coproduct ς n,1 * (1 −) :…”

“…In particular, if we take the groups {G n } n∈N to be the family of braid groups {B n } n∈N and the groups {H n } n∈N to be the family of free groups {F n } n∈N , Theorem A recovers [30,Theorem A]. Additionally, the families of symmetric groups or mapping class groups of orientable and non-orientable surfaces also fit into this framework (see Section 3).…”

Some computations of stable twisted homology for mapping class groups

“…Proposition 2.6. [28,Proposition 3.8] Let M be a pre-braided strict monoidal category and α : UQ −→ M be a strong monoidal functor. Then, the precomposition by α provides a functor P ol strong n…”