Foncteurs Faiblement Polynomiaux

Djament, Aurélien; Vespa, Christine

doi:10.1093/imrn/rnx099

Cited by 20 publications

(43 citation statements)

References 35 publications

(80 reference statements)

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“…Polynomial functors were originally defined using cross effects for functors with domain category a monoidal category with the unit 0 a null object (see [32]). Proposition 2.3 in [30] shows that strong polynomiality can also be defined in terms of cross effects, and hence that it is a direct generalisation of the classical notion of polynomiality. Note that if 0 is a null object in C, functors C → A are automatically split, as in that case we have a canonical factorisation id : A → X⊕A → A, natural in A.…”

Section: 2mentioning

confidence: 97%

Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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Abstract. Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups. By making systematic the proofs of classical stability results, we show that they all hold with the same type of coefficient systems, obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups, braid groups, automorphisms of free groups, unitary groups, mapping class groups of non-orientable surfaces and mapping class groups of 3-manifolds. Our construction can also be applied to families of groups not considered before in the context of homological stability.As a byproduct of our work, we construct the braided analogue of the category F I of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.A family of groupsis said to satisfy homological stability if the induced mapsare isomorphisms in a range 0 ≤ i ≤ f (n) increasing with n. In this paper, we prove that homological stability always holds if there is a monoidal category C satisfying a certain hypothesis and a pair of objects A and X in C, such that G n is the group of automorphisms of A ⊕ X ⊕n in C. We show that stability holds not just for constant coefficients, but also for both polynomial and abelian coefficients, without any further assumption on C. The polynomial coefficient systems considered here are functors F : C → Z -Mod satisfying a finite degree condition. They are generalisations of polynomial functors in the sense of functor homology, classically considered in homological stability, and include new examples such as the Burau representation of braid groups. Abelian coefficients are given by functors F : C → ZG ab ∞ -Mod for G ab ∞ the abelianisation of the limit group G ∞ , satisfying the same finite degree condition. These include coefficients such as the sign representation, or determinant-twisted polynomial functors. Such coefficient systems are newer to the subject. One consequence of stability with abelian coefficients is that stability with polynomial coefficients also holds (under the same conditions) for the commutator subgroups G n ≤ G n .Our theorem applies to all the classical examples and gives new stability results with twisted coefficients in particular for symmetric groups, alternating groups, unitary groups, braid groups, mapping class groups of non-orientable surfaces, automorphisms and symmetric automorphisms of free groups, and it proves stability for these groups...

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Section: 2mentioning

confidence: 97%

Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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“…Djament conjectured that the stable degree of H k (GL(R, I); k) is ≤ 2k in [Dja2, Conjecture 1] (also see [DV,§5.2] for further discussion). Part (1) of Application B proves that this stable degree is ≤ 2k + d. Thus, up to an additive constant, Application B establishes this conjecture.…”

Section: Introductionmentioning

confidence: 99%

Linear and quadratic ranges in representation stability

Church

Miller

Nagpal

et al. 2018

Advances in Mathematics

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We prove two general results concerning spectral sequences of FI-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FImodules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of Djament.

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“…This definition can also be applied to monoidal categories where the monoidal unit is a null object. Djament and Vespa introduce in [7] the definition of strong polynomial functors for symmetric monoidal categories with the monoidal unit being an initial object. Here, the category Uβ is neither symmetric, nor braided, but pre-braided in the sense of [20].…”

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confidence: 99%

“…Then, in Section 2, we introduce the Long-Moody functors, prove Theorem A and give some of their properties. In Section 3, we review the notion of strong polynomial functors and extend the framework of [7] to pre-braided monoidal categories. Finally, Section 4 is devoted to the proof of Theorem B and to some other properties of these functors.…”

mentioning

confidence: 99%

The Long–Moody construction and polynomial functors

Soulié¹

2019

Annales De l'Institut Fourier

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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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Foncteurs Faiblement Polynomiaux

Cited by 20 publications

References 35 publications

Homological stability for automorphism groups

Homological stability for automorphism groups

Linear and quadratic ranges in representation stability

The Long–Moody construction and polynomial functors

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