[Paper Review] Symmetric Powers and Norms of Mackey Functors

John Ullman|arXiv (Cornell University)|Apr 20, 2013

Homotopy and Cohomology in Algebraic Topology3 references3 citations

TL;DR

This paper provides algebraic descriptions of derived symmetric powers and norm constructions on Mackey functors using equivariant stable homotopy theory, establishing that the symmetric power of a Mackey functor corresponds to the degree-n part of the free Tambara functor on it. The key contribution is a new characterization of Tambara functors via multiplicative pushforwards, yielding a formula for the free Tambara functor that mirrors equivariant extended powers.

ABSTRACT

In this paper we give detailed algebraic descriptions of the derived symmetric power and norm constructions on categories of Mackey functors, as well as the derived G-symmetric monoidal structure. We build on the results of [Ull2], in which it is shown that every Tambara functor over a finite group G arises as the zeroth stable homotopy group of a commutative ring G-spectrum. The norm / restriction adjunctions on categories of Tambara functors promised in [Ull2] are demonstrated algebraically. Finally, we give a new characterization of Tambara functors in terms of multiplicative push forwards of Mackey functors, and use this to obtain an appealing new description of the free Tambara functor on a Mackey functor which closely matches the structure of equivariant extended powers.

Motivation & Objective

• To provide an algebraic description of the derived symmetric power construction on Mackey functors.
• To give an intrinsic algebraic formulation of the norm functor from H-Mackey functors to G-Mackey functors for subgroups H ≤ G.
• To define and describe a G-symmetric monoidal structure on Mackey functors via geometric fixed points.
• To re-derive the adjunction between restriction and norm functors on Tambara functors algebraically.
• To introduce multiplicative pushforwards of Mackey functors and use them to give a new characterization of Tambara functors as multiplicative Mackey functors.

Proposed method

• Use the derived free commutative ring G-spectrum functor 𝕮 and the π₀ functor to relate commutative ring G-spectra to Tambara functors.
• Define symmetric powers of Mackey functors as the n-th graded piece of the free Tambara functor on a given Mackey functor.
• Construct the norm functor N_H^G on Mackey functors via π₀ of the derived norm on Eilenberg-MacLane spectra.
• Use pullback diagrams in finite G-sets to describe structure maps in symmetric powers and norms.
• Introduce multiplicative pushforwards of Mackey functors to model multiplicative structure in Tambara functors.
• Establish isomorphisms between colimits over pullback diagrams and the free Tambara functor, matching the structure of equivariant extended powers.

Experimental results

Research questions

• RQ1 How can the derived symmetric power of a Mackey functor be described algebraically in terms of Tambara functors?
• RQ2 What is the algebraic structure of the norm functor N_H^G on Mackey functors for H ≤ G?
• RQ3 How does the G-symmetric monoidal structure on Mackey functors arise from geometric fixed points of spectra?
• RQ4 Can the adjunction between restriction and norm functors on Tambara functors be reconstructed algebraically from Mackey functor data?
• RQ5 What is the role of multiplicative pushforwards in characterizing Tambara functors as multiplicative Mackey functors?

Key findings

• The symmetric power Sym_n(ℳ) of a G-Mackey functor ℳ is isomorphic to the n-th graded piece of the free Tambara functor on ℳ.
• The norm functor N_H^G on Mackey functors is algebraically described as π₀ of the derived norm on Eilenberg-MacLane spectra, matching the left adjoint of restriction on Tambara functors.
• The G-symmetric monoidal structure on Mackey functors is described via geometric fixed points and pullback diagrams in finite G-sets.
• The free Tambara functor on a Mackey functor ℳ is isomorphic to the colimit over pullback diagrams in P(G;n), with structure maps induced by multiplicative pushforwards.
• Tambara functors are characterized as multiplicative Mackey functors via multiplicative pushforwards, providing a new algebraic model that mirrors equivariant extended power structures.
• The structure maps of the free Tambara functor are shown to coincide with those defined via colimits over pullback diagrams, confirming compatibility with equivariant topology.