Equivariant iterated loop space theory and permutative G-categories

TL;DR

This paper develops operadic foundations for equivariant iterated loop space theory, introduces genuine permutative G-categories, and applies these concepts to prove key theorems in equivariant stable homotopy theory and algebraic K-theory.

Contribution

It establishes a framework for equivariant iterated loop spaces, introduces genuine permutative G-categories, and provides new proofs of fundamental theorems like the equivariant Barratt-Priddy-Quillen and tom Dieck splitting.

Findings

Proved the equivariant Barratt-Priddy-Quillen theorem.

Provided a categorical proof of the tom Dieck splitting theorem.

Developed models for genuine G-spectra and E_{∞} G-categories.

Abstract

We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V-fold loop G-spaces to several avatars of a recognition principle for infinite loop G-spaces. We then explain what genuine permutative G-categories are and, more generally, what E_{\infty} G-categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt-Priddy-Quillen theorem as a statement about genuine G-spectra and use it to give a new, categorical, proof of the tom Dieck splitting theorem for suspension G-spectra. Other examples are geared towards equivariant algebraic K-theory.