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.
Cite
@article{arxiv.1207.3459,
title = {Equivariant iterated loop space theory and permutative G-categories},
author = {Bertrand Guillou and J. P. May},
journal= {arXiv preprint arXiv:1207.3459},
year = {2018}
}
Comments
63 pages. Final version, to appear in AGT. Section 1 is new and concerns E_V-operads and V-fold loop G-spaces, where V is a representation of G. Other sections have been reorganized. An appendix discusses pairings of operads, giving geometric examples with good behavior. The previous version mistakenly claimed that the Barrett-Eccles operad has a well-behaved self-pairing, and we explain the error