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" 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 Z max . 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 selfduality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

The cyclic and epicyclic sites

“…There are by now a number of equivalent descriptions of the cyclic and epicyclic categories, ranging from the most concrete i.e. given in terms of generators and relations, to the most conceptual as in [5]. The description of these categories in terms of oriented groupoïds turns out to be very useful to determine the points of the epicyclic topos by considering filtering colimits, in the category g, of the special points provided by the Yoneda embedding of the categories.…”

“…One has a canonical functor Mod ∶Λ op → N × which is trivial on the objects and associates to a semilinear map of semimodules over F = Z max the corresponding injective endomorphism Fr n ∈ End(F) (cf. [5] for details). This functor induces a geometric morphism of topoi Mod ∶ (Λ op ) ∧ → N × .…”

“…By Proposition 2.8 of [5], the epicyclic categoryΛ is canonically isomorphic to the full subcategory of Arc ⋉ N × whose objects are the archimedean sets m ∶= (Z, x ↦ x + m + 1) for m ≥ 0. The oriented groupoïd G(m) is by Lemma 2.4 canonically isomorphic to g(m).…”

“…The theory of topoi of Grothendieck provides the best geometric framework to understand cyclic homology and the λ-operations using the topos associated to the cyclic category [3] and its epicyclic refinement [5]. Given a small category C, we denote byĈ the topos of contravariant functors from C to the category of sets Sets.…”

“…Both the cyclic homology of R and its λ-operations are completely encoded by the associated sheaf R ♮ . In [5], we provided a conceptual understanding of the epicyclic category as projective geometry over the semifield F ∶= Z max of the tropical integers. In these terms the functorΛ → Fin considered above assigns to a projective space the underlying finite set.…”