Segal models for equivariant incomplete infinite loop spaces
Tjark Bantelmann
TL;DR
The paper develops equivariant incomplete infinite loop space machines by introducing Γ-spaces indexed by disk-like systems $\\mathcal{I}$ and compatible $G$-universes $U$, producing connective positive $\\Omega$-$G$-spectra. It unifies several equivariant Γ-space models (Gamma-G, Gamma_G-G, Gamma_I-G) and introduces two coherent notions of specialness adapted to incomplete data, along with transfer maps and comparison results. The main achievement is Theorem A, establishing an incomplete Segal machine $\\mathbb{S}^{G,U}_\mathcal{I}$ that converts $\\mathcal{I}$-special input into $\\Omega$-spectra, together with Theorem B, which constructs Segal $K$-theory spectra for $\\mathcal{I}$-normed permutative $G$-categories via an associated $\\Gamma_\mathcal{I}$-$G$-category. The framework clarifies how isotropy, transfers, and universe-completeness interact in the incomplete setting and provides a robust path to equivariant $K$-theory with normed structures, potentially guiding coefficient-system approaches for more general Mackey-functor phenomena. Overall, the work extends Segal-type infinite loop space machines to incomplete equivariant contexts, enabling new stable- and algebraic-structure constructions in equivariant homotopy theory.
Abstract
We model equivariant infinite loop spaces indexed on incomplete universes via suitable equivariant analogs of $Γ$-spaces. The choice of universe dictates a transfer system which in turn dictates the Segal condition on equivariant $Γ$-spaces. Equivariant $Γ$-spaces themselves come in different but equivalent guises interpolating between categories $Γ$ as defined by Segal and $Γ_G$ as defined by Shimakawa. The main application is the construction of Segal $K$-theory of normed permutative categories.