A symmetric monoidal and equivariant Segal infinite loop space machine

Abstract

In [12], we reworked and generalized equivariant infinite loop space theory, which shows how to construct G-spectra from G-spaces with suitable structure. In this paper, we construct a new variant of the equivariant Segal machine that starts from the category ▪ of finite sets rather than from the category ▪ of finite G-sets and which is equivalent to the machine studied in [19,12]. In contrast to the machine in [19,12], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of ▪–G-spaces to the symmetric monoidal category of orthogonal G-spectra. We relate it multiplicatively to suspension G-spectra and to Eilenberg–Mac Lane G-spectra via lax symmetric monoidal functors from based G-spaces and from abelian groups to ▪–G-spaces. Even non-equivariantly, this gives an appealing new variant of the Segal machine. This new variant makes the equivariant generalization of the theory essentially formal, hence likely to be applicable in other contexts.