[Paper Review] On Burnside Theory for groupoids

Laiachi El Kaoutit, Leonardo Spinosa|arXiv (Cornell University)|Jul 12, 2018

Homotopy and Cohomology in Algebraic Topology24 references4 citations

TL;DR

This paper extends Burnside theory from finite groups to groupoids by introducing conjugacy relations between subgroupoids and characterizing isomorphism of finite groupoid-sets via fixed-point counts under subgroupoids. The key contribution is a generalized Burnside Theorem for groupoids, establishing that two finite groupoid-sets are isomorphic if and only if they have identical fixed-point data across all single-object subgroupoids, with applications to the Burnside algebra and ghost map of finite groupoids.

ABSTRACT

We explore the concept of conjugation between subgroupoids, providing several characterizations of the conjugacy relation (Theorem A in §1.2). We show that two finite groupoid-sets, over a locally strongly finite groupoid, are isomorphic, if and only if, they have the same number of fixed points with respect to any subgroupoid with a single object (Theorem B in §1.2). Lastly, we examine the ghost map of a finite groupoid and the idempotents elements of its Burnside algebra. The exposition includes an Appendix where we gather the main general technical notions that are needed along the paper.

Motivation & Objective

• To generalize classical Burnside theory—originally formulated for finite groups—to the broader context of groupoids.
• To define and characterize conjugacy between subgroupoids in groupoids, providing multiple equivalent conditions for this relation.
• To establish a criterion for isomorphism of finite groupoid-sets based on fixed-point data under all single-object subgroupoids.
• To study the Burnside algebra of a finite groupoid, including its ghost map and the structure of its idempotent elements.
• To provide a categorical framework using Laplaza categories and Grothendieck functors to unify the construction of Burnside rings in groupoid settings.

Proposed method

• Introduces the monoidal category of groupoid-bisets and uses it to define induction functors and monoidal equivalences between groupoid-sets.
• Defines left and right cosets relative to subgroupoids and establishes the conjugacy equivalence relation on subgroupoids via these cosets.
• Develops the fixed-point functor for groupoid-sets and constructs the table of marks for finite groupoids as a matrix of equivariant map counts.
• Applies the Grothendieck functor to Laplaza categories to build the Burnside ring of a groupoid, ensuring compatibility with symmetric monoidal structures.
• Analyzes the ghost map from the Burnside algebra to the product of integers indexed by conjugacy classes of subgroupoids, and studies its kernel and image.
• Uses direct limits and product decompositions to analyze coproducts and products in the Burnside functor, particularly in the finite case.

Experimental results

Research questions

• RQ1 When are two finite groupoid-sets isomorphic, and what invariants fully classify them in the groupoid setting?
• RQ2 How can the conjugacy relation between subgroupoids in a groupoid be characterized algebraically and categorically?
• RQ3 What is the structure of the Burnside algebra of a finite groupoid, and how do its idempotent elements relate to the ghost map?
• RQ4 How does the ghost map of a finite groupoid reflect the fixed-point data of its groupoid-sets?
• RQ5 To what extent can the classical Burnside Theorem for groups be generalized to groupoids using fixed-point invariants?

Key findings

• Two finite groupoid-sets over a locally strongly finite groupoid are isomorphic if and only if they have the same number of fixed points with respect to every subgroupoid with a single object.
• The conjugacy relation between subgroupoids is characterized in multiple equivalent ways, including via coset bijections and equivariant isomorphisms of orbit sets.
• The ghost map of a finite groupoid is injective if and only if the fixed-point data uniquely determine the groupoid-set, generalizing the classical injectivity result for finite groups.
• The Burnside algebra of a finite groupoid admits a decomposition into a product of copies of Z indexed by conjugacy classes of subgroupoids, with the ghost map mapping each element to its fixed-point vector.
• The idempotent elements of the Burnside algebra correspond precisely to the characteristic functions of conjugacy classes of subgroupoids, under the ghost map.
• The Burnside functor preserves coproducts and products in the finite case, and the Burnside ring of a finite groupoid is isomorphic to the product of the Burnside rings of its connected components.