I'm trying to understand the relationship between cyclic spaces and S¹-equivariant homotopy theory. More precisely, I only care about S¹-spaces up to equivalence of fixed point spaces for the finite subgroups of S¹. Given a cyclic space X : ΔC^op → Top, I know the geometric realization of the restriction of X to Δ^op is an S¹-space. Form the associated fixed point diagram O^op → Spaces where O is the full subcategory of the orbit category of S¹ on the objects S¹/C where C ranges over finite subgroups of S¹. I regard the category of functors O^op → Spaces as an (∞,1)-category.

My question is, what structure on X does the resulting diagram depend on? More specifically, under what conditions does a map f : X → Y of cyclic spaces induce an equivalence of fixed point diagrams?

In O consider the full subcategory O₁ on the object S¹/{•}. The restriction of this diagram to O₁ is a space with S¹-action in the (∞,1)-categorical sense, and I think it's just the left Kan extension of X along the functor ΔC^op → BS¹ induced by the fact that ΔC is the quotient of something (ΔZ) by an S¹-action. Thus it only depends on X viewed as a functor from ΔC^op to the (∞,1)-category of spaces. But to evaluate on the other objects of O, corresponding to the fixed point spaces of nontrivial finite subgroups of S¹, do I need to know each X[r] as a C_r+1 space (i.e. the homotopy types of the fixed points sets for subgroups of C_r+1)? Is there a way to encode all of that information in a functor from some (maybe (∞,1)-)category to Spaces? Or is it possible that I need to remember even more information about X?