Foncteurs polynomiaux et foncteurs de Mackey non linéaires

Abstract

Résumé. -On décrit les foncteurs polynomiaux, des groupes abéliens libres vers les groupes abéliens, comme des diagrammes de groupes abéliens dont on explicite les relations. Abstract (Polynomial functors and nonlinear Mackey functors)Polynomial functors from free abelian groups to abelian groups are described explicitely in the form of diagrams of abelian groups, that are maps between the crosseffects of the polynomial functor which satisfy a list of relations. The key is to use an appropriate notion of Mackey functor from the category of finite sets and surjections.Soient Ab la catégorie des groupes abéliens, et F (Z) sa sous-catégorie pleine dont les objets sont les groupes abéliens libres de type fini. On sait que la caté-gorie des foncteurs additifs de F (Z) vers Ab estéquivalenteà la catégorie Ab.

Exactness of homotopy functors of spaces

“…This suggests looking for a natural construction of a spectral sequence whose E 2 term agrees with the E 2 term of the ad hoc spectral sequence. From that point, the generalization to n-exact functors is clear by using the extension of quadratic modules found in [4]. Such a spectral sequence does exist, and will be the subject of a future article.…”

Derived functors of the divided power functors

“…The notation ∂ Kos and the name 'Koszul kernel algebra' are justified by the fact (which will be proved in section 6.2) that the differential graded P Z -algebra (L * Γ Fp (A/p, 1), ∂ Kos ) is the tensor product of all the algebras Γ Fp A/p (r) [2] ⊗ Λ Fp A/p (r) [1] with a Koszul differential and of the algebra Λ Fp (A/p [1]) with the zero differential. We will prove in corollary 6.20 that the Koszul kernel algebra K F 2 (A/2) is very closely related to the following skew Koszul kernel algebra SK F 2 (A/2).…”

“…Proof. To prove (ii), we observe that by definition, ∂ must send the summand Γ 1 (A/p (r) [2]) into the homogeneous summand of degree 1 and weight p r , which is equal to Λ 1 (A/p (1) [1]). We now prove (i).…”

“…If we denote by π r , resp π ′ r , resp. π ′ 0 , the canonical projection of the right-hand side of isomorphism (6.9) onto the summands A/p (r) [2], A/p (r) [1], and A/p [1] respectively, then…”

“…. , r n ) with r i = 2, r k = 0 if k = i, (iv) n(n − 1)/2 generators of the form V (2) [n + j + 2i− 3], corresponding to the n-tuples (r 1 , . .…”