It is well known that very special –spaces and grouplike –spaces both model connective spectra. Both these models have equivariant analogues in the case when the group acting is finite. Shimakawa defined the category of equivariant –spaces and showed that special equivariant –spaces determine positive equivariant spectra. Costenoble and Waner [Trans. Amer. Math. Soc. 326 (1991) 485-505] showed that grouplike equivariant –spaces determine connective equivariant spectra.
We show that with suitable model category structures the category of equivariant –spaces is Quillen equivalent to the category of equivariant –spaces. We define the units of equivariant ring spectra in terms of equivariant –spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.
"Units of equivariant ring spectra." Algebr. Geom. Topol. 11 (3) 1361 - 1403, 2011. https://doi.org/10.2140/agt.2011.11.1361