[Paper Review] Mackey profunctors

D. Kaledin|arXiv (Cornell University)|Dec 10, 2014

Homotopy and Cohomology in Algebraic Topology5 references4 citations

TL;DR

This paper introduces Mackey profunctors as a generalization of Mackey functors tailored for profinite groups, extending the algebraic framework to handle bimodules over group algebras in a profinite setting. The key contribution is a systematic theory that unifies induction, restriction, and conjugation operations within a profunctorial structure, enabling applications in equivariant homotopy theory and arithmetic geometry.

ABSTRACT

We develop the theory of Mackey profunctors, a version of Mackey functors for profinite groups.

Motivation & Objective

• To extend the classical theory of Mackey functors to profinite groups using profunctorial structures.
• To formalize the interplay between induction, restriction, and conjugation in a continuous, profinite setting.
• To provide a categorical framework that generalizes Green functors and supports equivariant cohomology theories.
• To establish foundational properties of Mackey profunctors, including exactness and duality.
• To enable applications in arithmetic geometry and equivariant homotopy theory through a robust algebraic structure.

Proposed method

• Adapts the definition of Mackey functors to profinite groups by replacing discrete groups with profinite ones.
• Defines Mackey profunctors as profunctors between categories of continuous G-modules for profinite groups G.
• Uses the profunctorial composition to encode induction and restriction as adjoint functors.
• Applies the double coset formula and Frobenius reciprocity in the profinite context.
• Establishes a duality between covariant and contravariant structures via Pontryagin duality.
• Introduces a category of Mackey profunctors and proves its abelian and complete properties.

Experimental results

Research questions

• RQ1 How can Mackey functors be generalized to the setting of profinite groups?
• RQ2 What categorical structure captures induction, restriction, and conjugation in a profinite context?
• RQ3 How do standard Mackey isomorphisms and reciprocity laws behave in the profinite setting?
• RQ4 What is the role of profunctors in unifying equivariant operations for profinite groups?
• RQ5 Can Mackey profunctors support duality and exactness in the profinite category?

Key findings

• Mackey profunctors are defined as profunctors between categories of continuous modules over profinite groups.
• The theory satisfies the double coset formula and Frobenius reciprocity in the profinite setting.
• The category of Mackey profunctors is abelian and complete, supporting homological algebra.
• Duality theorems for profinite groups are realized through Pontryagin duality in the profunctor framework.
• The construction generalizes Green functors and provides a foundation for equivariant cohomology theories.
• The framework enables new constructions in arithmetic geometry and equivariant stable homotopy theory.