The cyclic and epicyclic sites

Abstract

We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of "max-plus integers" ${\mathbb{F}}_{\max }$. An object of this category is a pair $(E,K)$ of a semimodules $E$ over an algebraic extension $K$ of ${\mathbb{Z}}_{\max }$. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos $\widehat{{\mathbb{N}}^{\times }}$ of [6] and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations.