Projective geometry in characteristic one and the epicyclic category

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 of max-plus integers ${\mathbb{Z}}_{max}$. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ${\mathbb{Z}}_{max}$. The associated projective spaces are finite and provide a mathematically consistent interpretation of Tits’s 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.