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.

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

References in corpus (2)

• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Categorical models for equivariant classifying spaces

Cited by in corpus (11)

• Models of G-spectra as presheaves of spectra
• Combinatorial $N_{\infty }$ operads
• G-symmetric monoidal categories of modules over equivariant commutative ring spectra
• The 2-dimensional stable homotopy hypothesis
• Global algebraic K-theory
• Equivariant stable categories for incomplete systems of transfers
• Equivariant infinite loop space theory, the space level story
• The Equivariant Cobordism Category
• Operads in Unstable Global Homotopy Theory
• G-Global Homotopy Theory and Algebraic K-Theory
• A Thom Spectrum Model for $C_2$-Integral Brown--Gitler Spectra