[Paper Review] A Mackey-functor theoretic interpretation of biset functors

Hiroyuki Nakaoka|arXiv (Cornell University)|Nov 16, 2013

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper introduces a 2-category $Σ$ of finite sets with group actions, enabling biset functors to be interpreted as a special class of Mackey functors on $Σ$. By generalizing Dress's Mackey functor framework to include all finite group homomorphisms, the work establishes a categorical equivalence between biset functors and deflation-strict Mackey functors on $Σ$, providing a unified categorical foundation for global representation theory and biset functors.

ABSTRACT

In this article, we consider a formulation of biset functors using the 2-category of finite sets with variable finite group actions. We introduce a 2-category $\mathbb{S}$, on which a biset functor can be regarded as a special kind of Mackey functors. This gives an analog of Dress' definition of a Mackey functor, in the context of biset functors.

Motivation & Objective

• To extend the theory of Mackey functors beyond fixed groups to all finite groups and homomorphisms.
• To provide a categorical framework where biset functors naturally arise as a special class of Mackey functors.
• To generalize Dress's definition of Mackey functors to the context of biset functors using 2-categorical structures.
• To establish a natural equivalence between the category of biset functors and a subcategory of deflation-strict Mackey functors on a new 2-category $Σ$.

Proposed method

• Define a 2-category $Σ$ whose objects are finite sets with finite group actions, and whose 1-cells are equivariant maps.
• Introduce a category $Σ$-span and a span category construction to model biset operations via pullbacks and pushforwards.
• Construct a 2-category $Σ$ with bicoproducts and bipullbacks, enabling the definition of induction, restriction, and other operations as 1-cell compositions.
• Define Mackey functors on $Σ$ as functors preserving bicoproducts and bipullbacks, generalizing Dress’s original definition.
• Introduce the notion of 'deflation-strict' Mackey functors to capture the essential structure of biset functors.
• Prove that the category of biset functors is equivalent to the full subcategory of deflation-strict Mackey functors on $Σ$ via a natural isomorphism.

Experimental results

Research questions

• RQ1 How can biset functors be systematically interpreted as Mackey functors in a 2-categorical setting?
• RQ2 What structure must a 2-category possess to support a Mackey functor theory that generalizes both classical and global biset functors?
• RQ3 Can the full category of biset functors be embedded into a category of Mackey functors via a 2-categorical construction?
• RQ4 What role do bicoproducts and bipullbacks play in defining Mackey functors on a 2-category of group actions?
• RQ5 Is there a natural equivalence between biset functors and a distinguished class of Mackey functors on a suitably defined 2-category?

Key findings

• The 2-category $Σ$ of finite sets with finite group actions provides a natural categorical setting for defining Mackey functors that generalize classical and global theories.
• Biset functors are shown to be equivalent to deflation-strict Mackey functors on $Σ$, establishing a categorical unification.
• The construction of $Σ$ allows for a 2-categorical interpretation of elementary biset operations such as induction, restriction, inflation, and deflation.
• The paper proves that the assignment of a biset functor to its associated Mackey functor on $Σ$ yields a natural isomorphism, confirming full faithfulness and essential surjectivity.
• The stabilization of the span category of $Σ$-sets leads to a well-behaved category of Mackey functors that captures the full structure of biset functors.
• The equivalence between biset functors and deflation-strict Mackey functors on $Σ$ confirms that the 2-categorical framework is both sufficient and necessary for a global Mackey functor theory.