Cyclic structures and the topos of simplicial sets

Given a point p   of the topos Δˆ of simplicial sets and the corresponding flat covariant functor F:Δ⟶SetsF:Δ⟶Sets, we determine the extensions of FF to the cyclic category Λ⊃ΔΛ⊃Δ. We show that to each such cyclic structure on a point p   of Δˆ corresponds a group GpGp, that such groups can be noncommutative and that each GpGp is described as the quotient of a left-ordered group by the subgroup generated by a central element. Moreover for any cyclic set X the fiber (or geometric realization) of the underlying simplicial set of X at p   inherits canonically the structure of a GpGp-space. This gives a far reaching generalization of the well-known circle action on the geometric realization of cyclic sets.