Equivariant homotopy theory for pro-spectra

Abstract

We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G-homotopy theory is "pieced together" from the G/U-homotopy theories for suitable quotient groups G/U of G; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In the model category of equivariant spectra Postnikov towers are studied from a general perspective. We introduce pro-G-spectra and construct various model structures on them. A key property of the model structures is that pro-spectra are weakly equivalent to their Postnikov towers. We discuss two versions of a model structure with "underlying weak equivalences". One of the versions only makes sense for pro-spectra. In the end we use the theory to study homotopy fixed points of pro-G-spectra.

Cited by in corpus (10)

• Global homotopy theory
• Encoding Equivariant Commutativity via Operads
• Commutative ring objects in pro-categories and generalized Moore spectra
• Proper equivariant stable homotopy theory
• Stable finiteness properties of infinite discrete groups
• Delta-discrete $G$-spectra and iterated homotopy fixed points
• Rational Z_p-Equivariant Spectra
• Profinite and discrete G-spectra and iterated homotopy fixed points
• Homotopy theory of monoid actions via group actions and an Elmendorf style theorem
• Classifying rational G-spectra for profinite G