  We give a historical perspective on the role of the cyclic category in the
development of cyclic theory. This involves a continuous interplay between the
extension in characteristic one and in S-algebras, of the traditional
development of cyclic theory, and the geometry of the toposes associated with
several small categories involved. We clarify the link between various existing
presentations of the cyclic and the epicyclic categories and we exemplify the
role of the absolute coefficients by presenting the ring of the integers as
polynomials in powers of 3, with coefficients in the spherical group ring
S[C_2] of the cyclic group of order two. Finally, we introduce the pericyclic
category which unifies and refines two conflicting notions of epicyclic space
existing in the literature.
