The fundamental groupoid as a terminal costack

TL;DR

This paper demonstrates that for well-behaved topological spaces, the connected components and fundamental groupoid form the terminal cosheaf and costack, respectively, revealing a deep categorical structure in topology.

Contribution

It establishes that the assignment of the fundamental groupoid to open sets is the terminal costack for good topological spaces, linking algebraic topology with categorical structures.

Findings

The functor $U o ext{pi}_0(U)$ is the terminal cosheaf.

The functor $U o ext{Pi}_1(U)$ is the terminal costack.

Provides a categorical characterization of fundamental groupoid.

Abstract

Let $X$ be a topological space. We denote by $\pi_0(X)$ the set of connected components of $X$ and by $\Pi_1(U)$ the fundamental groupoid. In this paper we prove that for good topological spaces the assignments $U\mapsto\pi_0(U)$ and $U\mapsto\Pi_1(U)$ are the terminal cosheaf and costack respectively.