Cyclic theory and the pericyclic category

TL;DR

This paper explores the historical development of cyclic theory, clarifies the relationships between cyclic and epicyclic categories, and introduces the pericyclic category to unify existing concepts in the field.

Contribution

It provides a comprehensive overview of cyclic theory's evolution, clarifies the link between various cyclic categories, and introduces the pericyclic category as a unifying framework.

Findings

Clarifies the relationship between cyclic and epicyclic categories.

Introduces the pericyclic category to unify existing notions.

Demonstrates the role of absolute coefficients using the ring of integers.

Abstract

We give a historical perspective on the role of the cyclic category in the development of cyclic theory. This involves a continuous interplay between the extension in characteristic one and in S-algebras, of the traditional development of cyclic theory, and the geometry of the toposes associated with several small categories involved. We clarify the link between various existing presentations of the cyclic and the epicyclic categories and we exemplify the role of the absolute coefficients by presenting the ring of the integers as polynomials in powers of 3, with coefficients in the spherical group ring S[C_2] of the cyclic group of order two. Finally, we introduce the pericyclic category which unifies and refines two conflicting notions of epicyclic space existing in the literature.