Self-duality of the lattice of transfer systems via weak factorization systems

TL;DR

This paper establishes a self-duality in the lattice of transfer systems for finite groups by linking them to weak factorization systems, revealing new structural insights into $G$-spectra and operads.

Contribution

It introduces a novel correspondence between $G$-transfer systems and weak factorization systems, leading to a self-duality in their lattice structure for finite Abelian groups.

Findings

Demonstrates a correspondence for finite Abelian groups

Establishes a self-duality in the lattice of transfer systems

Links transfer systems to weak factorization systems

Abstract

For a finite group $G$, $G$-transfer systems are combinatorial objects which encode the homotopy category of $G$-$N_\infty$ operads, whose algebras in $G$-spectra are $E_\infty$ $G$-spectra with a specified collection of multiplicative norms. For $G$ finite Abelian, we demonstrate a correspondence between $G$-transfer systems and weak factorization systems on the poset category of subgroups of $G$. This induces a self-duality on the lattice of $G$-transfer systems.