Projective geometry in characteristic one and the epicyclic category

TL;DR

This paper reveals that cyclic and epicyclic categories, essential in cyclic homology and lambda operations, can be derived from projective geometry over the semifield of max-plus integers, linking algebraic and geometric concepts.

Contribution

It introduces a novel geometric framework in characteristic one that interprets cyclic categories through projective geometry over the max-plus semifield.

Findings

Cyclic and epicyclic categories originate from projective geometry in characteristic one.

Finite projective spaces over the semifield provide a geometric interpretation of Tits' idea.

The cyclic category's self-duality and permutation properties gain geometric meaning.

Abstract

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield F of "max-plus integers". Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of F. The associated projective spaces are finite and provide a mathematically consistent interpretation of J. Tits' original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.