arXiv subjects
Publications and source records attributed to Behrang Noohi.
These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.
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.
We study fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. Then use this additional structure to analyze geometric/ topological properties of stacks. We give an explicit formula for the fundamental group of the coarse moduli space. Also, we use these additional structures to give a necessary and sufficient for an algebraic stack to be uniformizable.