The fundamental group as the structure of a dually affine space

TL;DR

This paper explores the duality of affine spaces in algebraic geometry, demonstrating that fundamental groups of pointed topological spaces can be viewed as structures of dually affine spaces, with applications to Zariski closure operators.

Contribution

It introduces the dual of the Zariski closure operator and shows that fundamental groups naturally arise as structures of dually affine spaces, expanding the theoretical framework.

Findings

Fundamental groups are structures of dually affine spaces.

The dual of the Zariski closure operator is defined.

The 1-sphere and its copowers are complete objects under the Zariski dual closure operator.

Abstract

This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental groups are shown to be examples of complete objects with respect to the Zariski dual closure operator.