Foundations of Topological Stacks I

By B. Noohi. 2005

120 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b9a54400706e22064d46f31efe488e7d355ddad

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. B. Noohi
2. K. Behrend
3. Martin Ulirsch
4. Chenchang Zhu
5. Renzo Cavalieri
6. Jonathan Wise
7. P. Xu
8. Melody Chan

In Geometry and Topology

1. Chenchang Zhu
2. B. Noohi
3. K. Behrend
4. Martin Ulirsch
5. Renzo Cavalieri
6. P. Xu
7. Melody Chan
8. Jonathan Wise

In Mathematical Physics

1. Chenchang Zhu
2. B. Noohi
3. K. Behrend
4. P. Xu
5. Martin Ulirsch
6. Jonathan Wise
7. Renzo Cavalieri
8. Melody Chan

Citations in the corpus

• Uniformization of Deligne-Mumford curves (2005, methodology): “More casually speaking, this means that, for a smooth X, if we know the maps from smooth domains into X, then we know the maps from an arbitrary domain into X. We quote the following result from [ No1 ].”
• n-Groupoids and Stacky Groupoids (2008, background): “…where they are known as Delign–Munford (DM) stacks and Artin stacks in the étale and general cases respectively (see for example [39] for a good summary), and recently in the differential category by [6, 24, 31] and [27] (in the context of orbifolds) and the topological category by [17, 24, 29].”
• Differentiable stacks and gerbes (2006, background): “cal charts” on a differentiable stack. Establishing such a dictionary consists of the first part of the paper. We note that this viewpoint of connecting stacks with groupoids is somehow 2 folklore (see [15, 34, 39, 40]). However, we feel that it is useful to spell it out in detail in the differentiable”
• Homotopy types of topological stacks (2008, background): “In the case where X is Serre topological, it follows from Corollary 5.9 that the homotopy groups defined above are naturally isomorphic to the ones defined in ([ No1 ], §17).”
• A MODULI STACK OF TROPICAL CURVES (2017, background): “ces. There is a natural Grothendieck topology τopen on Top, whose coverings consists of coverings by open subsets. In order to define the notion of a topological stack there are many choices for P(see [Noo05], [Met03], and [BGNX12] for details). Example 1.7. Let k be a non-Archimedean field and let C ”