A symmetric monoidal and equivariant Segal infinite loop space machine

TL;DR

This paper introduces a new symmetric monoidal equivariant Segal infinite loop space machine based on finite sets, providing a flexible, formal approach to constructing $G$-spectra with potential applications beyond the equivariant setting.

Contribution

It develops a novel variant of the equivariant Segal machine using finite sets, establishing a lax symmetric monoidal functor to orthogonal $G$-spectra, and simplifies the equivariant generalization of the theory.

Findings

Constructs a new equivariant Segal machine from finite sets.

Establishes a lax symmetric monoidal functor to orthogonal $G$-spectra.

Relates to suspension and Eilenberg-MacLane $G$-spectra.

Abstract

In [MMO] (arXiv:1704.03413), 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 $\scr{F}$ of finite sets rather than from the category ${\scr{F}}_G$ of finite $G$-sets and which is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of $\scr{F}$-$G$-spaces to the symmetric monoidal category of orthogonal $G$-spectra. We relate it multiplicatively to suspension $G$-spectra and to Eilenberg-MacLane $G$-spectra via lax symmetric monoidal functors from based $G$-spaces and from abelian groups to $\scr{F}$-$G$-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 is likely to be applicable in other contexts.