Equivariant iterated loop space theory and permutative $G$–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.