Equivariant infinite loop space theory, the space level story

Abstract

We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\Omega$-$G$-spectra for any topological group $G$, but our focus is on the construction of genuine $\Omega$-$G$-spectra when $G$ is finite. We also show what is and is not true when $G$ is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$ but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.

Fixed half a page of missing text on page 22 (which was cut due to a previous copy and paste error)

References in corpus (11)

• On an extension of the notion of Reedy category
• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Six model structures for DG-modules over DGAs: Model category theory in homological action
• On the construction of functorial factorizations for model categories
• Continuous functors as a model for the equivariant stable homotopy category
• The theory and practice of Reedy categories
• Equivariant iterated loop space theory and permutative G-categories
• Units of equivariant ring spectra
• Enriched model categories and presheaf categories
• A model structure on internal categories in simplicial sets
• Categorical models for equivariant classifying spaces

Cited by in corpus (3)

• A multiplicative comparison of Waldhausen and Segal K-theory
• Loday constructions of Tambara functors
• Partial parametrized presentability and the universal property of equivariant spectra