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 : &Delta;C<sup>op</sup> &rarr; Top, I know the geometric realization of the restriction of X to &Delta;<sup>op</sup> is an S<sup>1</sup>-space.  Form the associated fixed point diagram <b>O</b><sup>op</sup> &rarr; 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> &rarr; Spaces as an (&infin;,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 &rarr; 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>/{&bull;}.  The restriction of this diagram to <b>O</b><sub>1</sub> is a space with S<sup>1</sup>-action in the (&infin;,1)-categorical sense, and I think it's just the left Kan extension of X along the functor &Delta;C<sup>op</sup> &rarr; BS<sup>1</sup> induced by the fact that &Delta;C is the quotient of something (&Delta;Z) by an S<sup>1</sup>-action.  Thus it only depends on X viewed as a functor from &Delta;C<sup>op</sup> to the (&infin;,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 (&infin;,1)-)category to Spaces?  Or is it possible that I need to remember even more information about X?