Cyclic structures and the topos of simplicial sets

TL;DR

This paper explores the extension of flat covariant functors from the simplicial to the cyclic category, revealing that each point in the topos corresponds to a potentially noncommutative group acting on geometric realizations.

Contribution

It characterizes cyclic structures on points of the topos of simplicial sets and relates them to specific groups, generalizing circle actions on cyclic set realizations.

Findings

Each cyclic structure corresponds to a group G(p).

Groups G(p) can be noncommutative and are quotients of left-ordered groups.

The geometric realization of cyclic sets inherits a G(p)-space structure.

Abstract

Given a point p of the topos of simplicial sets and the corresponding flat covariant functor F from the small category Delta to the category of sets, we determine the extensions of F to the cyclic category. We show that to each such cyclic structure on a point p of the topos of simplicial sets corresponds a group G(p), that such groups can be noncommutative and that each G(p) is described as the quotient of a left-ordered group by the subgroup generated by a central element. Moreover for any cyclic set X, the fiber (or geometric realization) of the underlying simplicial set of X at p inherits canonically the structure of a G(p)-space. This gives a far reaching generalization of the well-known circle action on the geometric realization of cyclic sets.