Foundations of Topological Stacks I

TL;DR

This paper develops a homotopy theory for topological stacks, introducing homotopy groups, Galois theory of coverings, and characterizations of quotient stacks, bridging algebraic and topological perspectives.

Contribution

It establishes foundational concepts like homotopy groups and Galois theory for topological stacks, and connects algebraic stacks with topological stacks.

Findings

Defined homotopy groups for topological stacks

Developed Galois theory of covering spaces for stacks

Characterized topological stacks as quotients of spaces by group actions

Abstract

This is the first in a series of papers devoted to foundations of topological stacks. We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of topological spaces. In this paper we go as far as introducing the homotopy groups and establishing their basic properties. We also develop a Galois theory of covering spaces for a (locally connected semilocally 1-connected) topological stack. Built into the Galois theory is a method for determining the stacky structure (i.e., inertia groups) of covering stacks. As a consequence, we get for free a characterization of topological stacks that are quotients of topological spaces by discrete group actions. For example, this give a handy characterization of good orbifolds. Orbifolds, graphs of groups, and complexes of groups are examples of topological (Deligne-Mumford) stacks. We also show that any algebraic stack (of finite type over $\mathbb{C}$) gives rise to a topological stack. We also prove a Riemann Existence Theorem for stacks. In particular, the algebraic fundamental group of an algebraic stack over $\mathbb{C}$ is isomorphic to the profinite completion of the fundamental group of its underlying topological stack. The next paper in the series concerns function stacks (in particular loop stacks) and fibrations of topological stacks. This is the first in a series of papers devoted to foundations of topological stacks.