I'm trying to understand the relationship between cyclic spaces and S^{1}-equivariant homotopy theory. More precisely, I only care about S^{1}-spaces up to equivalence of fixed point spaces for the finite subgroups of S^{1}. Given a cyclic space X : ΔC^{op} → Top, I know the geometric realization of the restriction of X to Δ^{op} is an S^{1}-space. Form the associated fixed point diagram **O**^{op} → Spaces where **O** is the full subcategory of the orbit category of S^{1} on the objects S^{1}/C where C ranges over finite subgroups of S^{1}. 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 **O**_{1} on the object S^{1}/{•}. The restriction of this diagram to **O**_{1} is a space with S^{1}-action in the (∞,1)-categorical sense, and I think it's just the left Kan extension of X along the functor ΔC^{op} → BS^{1} induced by the fact that ΔC is the quotient of something (ΔZ) by an S^{1}-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 S^{1}, 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?

**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^{op} → 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^{1}-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.