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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 : ΔCop → Top, I know the geometric realization of the restriction of X to Δop is an S1-space. Form the associated fixed point diagram Oop → 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 Oop → 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 ΔCop → 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 ΔCop 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 Cr+1 space (i.e. the homotopy types of the fixed points sets for subgroups of Cr+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 ΔCop Top, such that the identity functor to the injective model structure is a left Quillen functor, and such that the geometric realization to genuine S1-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.

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# Cyclic spaces and S^1-equivariant homotopy theory

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 : ΔCop → Top, I know the geometric realization of the restriction of X to Δop is an S1-space. Form the associated fixed point diagram Oop → 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 Oop → 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 ΔCop → 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 ΔCop 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 Cr+1 space (i.e. the homotopy types of the fixed points sets for subgroups of Cr+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?