Mackey functors and bisets

For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.     

Tambara Functors on Profinite Groups and Generalized Burnside Functors

The Tambara functor was defined by Tambara in the name of TNR-functor, to treat certain ring-valued Mackey functors on a finite group. Recently Brun revealed the importance of Tambara functors in the Witt–Burnside construction. In this article, we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshida's generalized Burnside ring functor is the first example. Consequently, we can consider a Tambara functor on any profinite group. In relation with the Witt–Burnside construction, we can give a Tambara-functor structure on Elliott's functor V _M , which generalizes the completed Burnside ring functor of Dress and Siebeneicher.     

Tambarization of a Mackey functor and its application to the Witt–Burnside construction

For an arbitrary group G, a (semi-)Mackey functor is a pair of covariant and contravariant functors from the category of G-sets, and is regarded as a G-bivariant analog of a commutative (semi-)group. In this view, a G-bivariant analog of a (semi-)ring should be a (semi-)Tambara functor. A Tambara functor is firstly defined by Tambara, which he called a TNR-functor, when G is finite. As shown by Brun, a Tambara functor plays a natural role in the Witt–Burnside construction.It will be a natural question if there exist sufficiently many examples of Tambara functors, compared to the wide range of Mackey functors. In the first part of this article, we give a general construction of a Tambara functor from any Mackey functor, on an arbitrary group G. In fact, we construct a functor from the category of semi-Mackey functors to the category of Tambara functors. This functor gives a left adjoint to the forgetful functor, and can be regarded as a G-bivariant analog of the monoid-ring functor.In the latter part, when G is finite, we investigate relations with other Mackey-functorial constructions — crossed Burnside ring, Elliott?s ring of G-strings, Jacobson?s F-Burnside ring — all these lead to the study of the Witt–Burnside construction.     

An equivariant tensor product on Mackey functors

For all subgroups H of a cyclic p-group G we define norm functors that build a G-Mackey functor from an H-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.     

Biset Functors as Module Mackey Functors and its Relation to Derivators

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite categories.     

SHAPE AND INDUCED REPRESENTATIONS

Abstract

Let K: P → T be a fixed functor. A criterion is given for a functor M': T → V to be a (right) Kan extension along K of some functor M: P → V. The functors M having a given M' as Kan extension are, in general, classified by continuous functors (V ^P)^o → V. We introduce a notion of system of imprimitivity, generalizing that of Mackey; when the shape category of K is codense in the systems of imprimitivity classify the functors H having M' as Kan extension. As a special case one obtains Mackey's Imprimitivity Theorem for finite groups.     

Stable equivariant abelianization, its properties, and applications

Let G be a finite group. For a based G-space X and a Mackey functor M, a topological Mackey functor is constructed, which will be called the stable equivariant abelianization of X with coefficients in M. When X is a based G-CW complex, is shown to be an infinite loop space in the sense of G-spaces. This gives a version of the RO(G)-graded equivariant Dold-Thom theorem. Applying a variant of Elmendorf's construction, we get a model for the Eilenberg-Mac Lane spectrum HM. The proof uses a structure theorem for Mackey functors and our previous results.     

Equivariant homotopical homology with coefficients in a Mackey functor

Let M be a Mackey functor for a finite group G. In this paper, generalizing the Dold-Thom construction, we construct an ordinary equivariant homotopical homology theory with coefficients in M, whose values on the category of finite G-sets realize the bifunctor M, both covariantly and contravariantly. Furthermore, we extend the contravariant functor to define a transfer in the theory for G-equivariant covering maps. This transfer is given by a continuous homomorphism between topological abelian groups.We prove a formula for the composite of the transfer and the projection of a G-equivariant covering map and characterize those Mackey functors M for which that formula has an expression analogous to the classical one.     

Rational Mackey functors for compact lie groups I

A Mackey functor M is a structure analogous to the representationring functor H R(H) encoding good formal behaviour under inductionand restriction. More explicitly, M associates an abelian groupM(H) to each closed subgroup H of a fixed compact Lie groupG, and to each inclusion K H it associates a restriction map and an induction map . This paper gives an analysis of thecategory of Mackey functors M whose values are rational vectorspaces: such a Mackey functor may be specified by giving a suitablycontinuous family consisting of a Q ₀(W_G(H))-module V(H) foreach closed subgroup H with restriction maps V(K) V(K) wheneverK is normal in K and K/K is a torus (a ‘continuous Weyl-toralmodule’). We show that the category of rational Mackeyfunctors is equivalent to the category of rational continuousWeyl-toral modules. In Part II this will be used to give analgebraic analysis of the category of rational Mackey functors,showing in particular that it has homological dimension equalto the rank of the group. 1991 Mathematics Subject Classification:19A22, 20C99, 22E15, 55N91, 55P42, 55P91.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

Induction theorems of surgery obstruction groups

Let be a finite group. It is well known that a Mackey functor is a module over the Burnside ring functor , where ranges over the set of all subgroups of . For a fixed homomorphism , the Wall group functor is not a Mackey functor if is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor . In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.

     

On the Projective Dimensions of Mackey Functors

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedekind domain. We conclude by completing work of Arnold on the global dimension of cohomological Mackey functors over ?.     

Extension groups between simple Mackey functors

In order to better understand the structure of indecomposable projective Mackey functors, we study extension groups of degree 1 between simple Mackey functors. We explicitly determine these groups between simple functors indexed by distinct normal subgroups. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups. We then focus on the case where the simple Mackey functors are indexed by the same subgroup. In this case, the corresponding extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine all extension groups between simple Mackey functors for a p-group and for a group that has a normal p-Sylow subgroup. Finally, we compute higher extension groups between simple Mackey functors for a group that has a p-Sylow subgroup of order p.     

Complexity and cohomology of cohomological Mackey functors

Let k be a field of characteristic p>0. Call a finite group G a poco group over k if any finitely generated cohomological Mackey functor for G over k has polynomial growth. The main result of this paper is that G is a poco group over k if and only if the Sylow p-subgroups of G are cyclic, when p>2, or have sectional rank at most 2, when p=2.A major step in the proof is the case where G is an elementary abelian p-group. In particular, when p=2, all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra of self extensions of the simple functor , by explicit generators and relations.     

Mackey structures on equivariant algebraic K-theory

We study the Mackey structure of the G-spectrum K _G C associated to a monoidal G-category C. It is proved that the coefficient system of K _G C coincides, as a (graded) Mackey functor, with the system of equivariant K-groups in the sense of Fröhlich and Wall. It is also shown that for any exact category U, there exists a G-spectrum Q _G U representing the equivariant K-theory of U in the sense of Dress and Kuku, and that Q _G U is naturally G-homotopy equivalent to K _G IsoU if every short exact sequence in U splits.     

Stratifications and Mackey Functors I: Functors For a Single Group

In the context of Mackey functors we introduce a category whichis analogous to the category of modules for a quasi-hereditaryalgebra which have a filtration by standard objects. Many ofthe constructions which work for quasi-hereditary algebras canbe done in this new context. In particular, we construct ananalogue of the ‘Ringel dual’, which turns out hereto be a standardly stratified algebra. The Mackey functors whichplay the role of the standard objects are constructed in thesame way as functors which have been used previously in parametrizingthe simple Mackey functors, but instead of using simple modulesin their construction (as was done before) we use p-permutationmodules. These Mackey functors are obtained as adjoints of theoperations of forming the Brauer quotient and its dual. Thefiltrations which have these Mackey functors as their factorsare closely related to the filtrations whose terms are the sumof induction maps from specified subgroups, or are the commonkernel of restriction maps to these subgroups. These latterfiltrations appear in Conlon's decomposition theorems for theGreen ring, as well as in other places, where they arise quitenaturally. 2000 Mathematics Subject Classification: primary 20C20; secondary20J05, 19A22, 16G70, 16E60.     

Flows With Respect to a Functor

Flows with respect to a functor F are introduced as a common generalization of the concepts of F-co-structured sinks and small F-co-structured sources. Appropriate factorization structures for functors are investigated and used to obtain several results that characterize coadjoint functors that have domains with various completeness conditions. When the functor in question is an identity functor, these results reduce to earlier results of Herrlich and Meyer for flows in a category. Functors of the type in question are shown to be nicely behaved with respect to composition. The dual notion of wolfs with respect to a functor is introduced, as is the concept of (co)limit with respect to a functor.     

Mackey first countability and docile locally convex spaces

We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding arbitrary quotients of Mackey first countable spaces. Some applications of the main result to spaces such as inductive limits are also given.