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In [MMO] (arXiv:1704.03413), we reworked and generalized equivariant infiniteloop space theory, which shows how to construct $G$-spectra from $G$-spaceswith suitable structure. In this paper, we construct a new variant of theequivariant Segal machine that starts from the category $\scr{F}$ of finitesets rather than from the category ${\scr{F}}_G$ of finite $G$-sets and whichis equivalent to the machine studied by Shimakawa and in [MMO]. In contrast tothe machine in [MMO], the new machine gives a lax symmetric monoidal functorfrom the symmetric monoidal category of $\scr{F}$-$G$-spaces to the symmetricmonoidal category of orthogonal $G$-spectra. We relate it multiplicatively tosuspension $G$-spectra and to Eilenberg-MacLane $G$-spectra via lax symmetricmonoidal functors from based $G$-spaces and from abelian groups to$\scr{F}$-$G$-spaces. Even non-equivariantly, this gives an appealing newvariant of the Segal machine. This new variant makes the equivariantgeneralization of the theory essentially formal, hence is likely to beapplicable in other contexts.

A symmetric monoidal and equivariant Segal infinite loop space machine