The Cyclic and Epicyclic Sites

TL;DR

This paper characterizes the points of the epicyclic topos, linking it to projective geometry over characteristic one and elucidating its role in cyclic homology and lambda operations.

Contribution

It establishes an equivalence between the points of the epicyclic topos and projective geometry in characteristic one, revealing new geometric insights into cyclic homology.

Findings

Points of the epicyclic topos correspond to projective geometry in characteristic one.

The category of points is equivalent to pairs of algebraic extensions and semimodules.

The epicyclic topos relates to the arithmetic topos and cyclic site.

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. An object of this category is a pair of an algebraic extension of the semifield and an archimedean semimodule over this extension. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos which we recently introduced 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.