A discrete model of S1S1-homotopy theory

We construct a discrete model of the homotopy theory of S1S1-spaces. We define a category PP with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. PP inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between PP and the model category of S1S1-spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets.