Generalized Long–Moody functors

Abstract

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.

The Burau representations of loop braid groups

“…In this section, we review the notions and basic properties of strong, very strong, split and weak polynomial functors. The definitions and results extend verbatim to the present slightly larger framework from the previous literature on that topic (see [DV19] and [Sou22,§4] for instance), the various proofs being mutatis mutandis generalisations of these previous works. For the remainder of §3.1, we fix a Grothendieck category A, a left-module (M, ) over strict monoidal small groupoid (G, , 0), where M is small groupoid, (G, , 0) has no zero divisors and Aut G (0) = {id 0 }.…”

“…Furthermore, the notion of weak polynomial functor is first introduced in [DV19, §1] and reflects more accurately the stable behaviour of the objects of the category Fct(M, A); see [DV19,§5] and Djament [Dja17]. The notions of strong and weak polynomial functors are then extended in [Sou22,§4] to the larger setting where M is a full subcategory of a pre-braided monoidal category where the unit is an initial object. Also the notion of very strong polynomial functor in this context is introduced there: it is closely related to the notion of coefficient systems of finite degree of Randal-Williams and Wahl [RW17].…”

“…The following properties are proven in [Sou22,Props. 4.4,4.10] (split polynomial functors are not considered there, but their study follows repeating mutatis mutandis this reference).…”

“…Let K( G, M , A) be the full subcategory of Fct( G, M , A) of the objects F such that κ(F ) = F . The category K( G, M , A) is a thick subcategory of Fct( G, M , A) and it is closed under colimits; see[Sou22, Prop. 4.6].…”

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

“…By Proposition 2.2, the first cohomology groups of the unit tangent bundle of the surfaces define a contravariant system H∨ : UM op 2 → Ab. A natural notion of polynomiality may be defined over the category UM op 2 , following for instance the analogous opposite notions of [DV19, §2] or[Sou22, §4]. However, as far as the authors know, this notion has no application for the questions addressed in this paper; see §2.2.2.…”