I'm trying to understand the relationship between cyclic spaces and S<sup>1</sup>-equivariant homotopy theory. More precisely, I only care about S<sup>1</sup>-spaces up to equivalence of fixed point spaces for the finite subgroups of S<sup>1</sup>. Given a cyclic space X : ΔC<sup>op</sup> → Top, I know the geometric realization of the restriction of X to Δ<sup>op</sup> is an S<sup>1</sup>-space. Form the associated fixed point diagram <b>O</b><sup>op</sup> → Spaces where <b>O</b> is the full subcategory of the orbit category of S<sup>1</sup> on the objects S<sup>1</sup>/C where C ranges over finite subgroups of S<sup>1</sup>. I regard the category of functors <b>O</b><sup>op</sup> → 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 <b>O</b> consider the full subcategory <b>O</b><sub>1</sub> on the object S<sup>1</sup>/{•}. The restriction of this diagram to <b>O</b><sub>1</sub> is a space with S<sup>1</sup>-action in the (∞,1)-categorical sense, and I think it's just the left Kan extension of X along the functor ΔC<sup>op</sup> → BS<sup>1</sup> induced by the fact that ΔC is the quotient of something (ΔZ) by an S<sup>1</sup>-action. Thus it only depends on X viewed as a functor from ΔC<sup>op</sup> to the (∞,1)-category of spaces. But to evaluate on the other objects of <b>O</b>, corresponding to the fixed point spaces of nontrivial finite subgroups of S<sup>1</sup>, do I need to know each X[r] as a C<sub>r+1</sub> space (i.e. the homotopy types of the fixed points sets for subgroups of C<sub>r+1</sub>)? 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?
**Edit:** I guess another way to phrase the question is this: I'm looking for a model category structure on the category of functors ΔC<sup>op</sup> → Top, such that the identity functor to the injective model structure is a left Quillen functor, and such that the geometric realization to genuine S<sup>1</sup>-spaces is also a left Quillen functor. Furthermore I would like to know whether this model category structure is Quillen equivalent to a diagram category of spaces (possibly on a topological index category) with objectwise weak equivalences.