Global orthogonal spectra

Abstract

For any compact Lie group $G$, there are several well-established definitions of a $G$-equivariant spectrum. In this paper, we develop the definition of a global orthogonal spectrum. Loosely speaking, this is a coherent choice of orthogonal $G$-spectrum for each compact Lie group $G$. We use the framework of enriched indexed categories to make this precise. We also consider equivariant $K$-theory and $\operatorname{Spin}^c$-cobordism from this perspective, and we show that the Atiyah-Bott-Shapiro orientation extends to the global context.