## 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_{∞}-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.

Original language | English |
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Pages (from-to) | 3259-3339 |

Number of pages | 81 |

Journal | Algebraic and Geometric Topology |

Volume | 17 |

Issue number | 6 |

DOIs | |

State | Published - Oct 4 2017 |

### Bibliographical note

Publisher Copyright:© 2017, Mathematical Sciences Publishers. All rights reserved.

## Keywords

- Equivariant algebraic K-theory
- Equivariant infinite loop spaces
- Permutative categories

## ASJC Scopus subject areas

- Geometry and Topology