On the geometric realization and subdivisions of dihedral sets

TL;DR

This paper reviews Drinfeld's simplified approach to geometric realization of simplicial and cyclic sets, extends it to dihedral sets, and clarifies the subdivisions of these structures.

Contribution

It introduces an analogous expression for dihedral sets' geometric realization and clarifies the subdivisions of simplicial, cyclic, and dihedral sets.

Findings

Expresses geometric realization of dihedral sets as filtered colimits.

Provides a clearer description of subdivisions for these sets.

Extends Drinfeld's method to dihedral sets.

Abstract

By expressing the geometric realization of simplicial sets and cyclic sets as filtered colimits, Drinfeld (arXiv:math/0304064v3) proved in a substantially simplified way the fundamental facts that geometric realization preserves finite limits, and that the group of orientation-preserving homeomorphisms of the interval [0,1] (resp. the circle R/Z) acts on the realization of a simplicial (resp. cyclic) set. In this paper, we first review Drinfeld's method and then introduce an analogous expression for the geometric realization of dihedral sets. We also see how these expressions lead to a clarified description of subdivisions of simplicial, cyclic, and dihedral sets.