Abstract:
The main result of the paper is a formula for the fundamental group of the coarse moduli space of a topological stack. As an application, we find simple general formulas for the fundamental group of the coarse quotient of a group action on a topological space in terms of the fixed point data. The formulas seem, surprisingly, to be new. In particular, we recover, and vastly generalize, results of Armstrong, Bass, Higgins, Rhodes.

Abstract:
In this note we define fibrations of topological stacks and establish their main properties. We prove various standard results about fibrations (fiber homotopy exact sequence, Leray-Serre and Eilenberg-Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. When restricted to topological spaces our notion of fibration coincides with the classical one.

Abstract:
We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem which shows that the weak homotopy type of the mapping stack Map(Y,X) does not change if we replace X by its classifying space, provided that Y is paracompact topological space. As an example, we describe the loop stack of the classifying stack BG of a topological group G in terms of twisted loop groups of G.

Abstract:
We compare three different ways of defining group cohomology with coefficients in a crossed-module: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossed-module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed-modules and also prove the "long" exact cohomology sequence associated to a short exact sequence of crossed-modules and weak morphisms.

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.

Abstract:
We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2-category of pointed homotopy 2-types. We indicate how certain standard notions of 2-group theory (e.g., kernels, cokernels, extension of 2-groups, and so on) find a simple description in terms of butterflies. We also discuss braided and abelian butterflies.

Abstract:
We introduce an explicit method for studying actions of a group stack G on an algebraic stack X. As an example, we study in detail the case where X=P(n_0,...,n_r) is a weighted projective stack over an arbitrary base S. To this end, we give an explicit description of the group stack of automorphisms of, the weighted projective general linear 2-group PGL(n_0,...,n_r). As an application, we use a result of Colliot-Thelene to show that for every linear algebraic group G over an arbitrary base field k (assumed to be reductive if char(k)>0) such that Pic}(G)=0, every action of G on P(n_0,...,n_r) lifts to a linear action of G on A^{r+1}.

Abstract:
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.