Foundations of Topological Stacks I
Foundations of Topological Stacks I
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拓扑堆栈基础 I
DOI：
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发表时间：
2005
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通讯作者：
B. Noohi
中科院分区：
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作者：
B. Noohi
这是专门讨论拓扑堆栈基础的系列论文中的第一篇。
我们开始沿着拓扑空间的经典同伦理论发展拓扑堆栈的同伦理论。在本文中，我们甚至介绍了同伦群并建立了它们的基本性质。我们还开发了覆盖（局部连通半局部 1-连通）拓扑堆栈空间的伽罗瓦理论。伽罗瓦理论内置了一种确定覆盖堆栈的堆栈结构（即惯性群）的方法。因此，我们免费获得了拓扑堆栈的表征，拓扑堆栈是离散群动作对拓扑空间的商。例如，这可以方便地描述良好的轨道折叠。
轨道折叠、群图和群复合体是拓扑 (Deligne-Mumford) 堆栈的示例。我们还证明任何代数栈（$mathbb{C}$ 上的有限类型）都会产生拓扑栈。我们还证明了栈的黎曼存在定理。特别是，$mathbb{C}$ 上的代数栈的代数基本群与其基础拓扑栈的基本群的有限完备同构。
该系列的下一篇论文涉及函数堆栈（特别是循环堆栈）和拓扑堆栈的纤维化。这是专门讨论拓扑堆栈基础的系列论文中的第一篇。
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.