The \'Etale Fundamental Groupoid as a Terminal Costack

TL;DR

This paper proves that the assignment of the étale fundamental groupoid to open subschemes of a noetherian scheme forms the 2-terminal object in the category of costacks over the étale site, establishing a universal property.

Contribution

It establishes that the étale fundamental groupoid assignment is the 2-terminal costack over the étale site of a noetherian scheme, providing a universal characterization.

Findings

The assignment $U o \\Pi_1(U)$ is the 2-terminal costack.

This provides a universal property for the étale fundamental groupoid.

The result applies to noetherian schemes and their étale coverings.

Abstract

Let $X$ be a noetherian scheme. We denote by $\Pi_1(X)$ the fundamental groupoid. In this paper we prove that the assignments $U\mapsto\Pi_1(U)$ is the 2-terminal costack over the site of \'etale coverings of $X$.