  We construct a discrete model of the homotopy theory of $S^1$-spaces. We
define a category $\sP$ with objects composed of a simplicial set and a cyclic
set along with suitable compatibility data. $\sP$ 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 $\sP$ and the model
category of $S^1$-spaces in which weak equivalences and fibrations are maps
inducing weak equivalences and fibrations on passage to all fixed point sets.
