"Fundamental groups of algebraic stacks." This has appeared in J. Inst. Math. Jussieu, 3 (2004), no. 1, 69--103 (pdf).
Abstract: This paper generalizes the results of a part of my M.I.T. thesis. It is shown that the category of representable finite etale covers of an algebraic stack is a Galois category. The associated fundamental group generalizes the corresponding notion for schemes, but is equipped with an additional structure coming from inertia groups. This additional structure governs the stacky structure of the covering stacks and, in particular, is used to give a very simple and handy criterion for an algebraic stack to be uniformizable (i.e. quotient of an algebraic space by a finite group action). We also use this additional structure to give a simple formula for the fundamental group of the coarse moduli space. As in application, we obtain a formula for the fundamental group of the coarse quotient of a group scheme acting on a scheme.
"Uniformization of Deligne-Mumford analytic curves," with Kai Behrend. This has appeared in J. Reine Angew. Math., 599 (2006), 111--153 (pdf).
Abstract: We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary Deligne-Mumford curve as a quotient stack. A major player in the game is the homotopy theory of 2-groups (=crossed-modules). We also discuss connections with the theory of F-groups and Bass-Serre theory of graphs of groups.
" Foundations of topological stacks I." Preprint, 81 pages, arXiv:math/0503247v1 [math.AG] (pdf).
Abstract: This is the first in a series of notes on topological stacks. It covers the basic definitions, homotopy groups, covering spaces, relations with algebraic and analytic stacks, Riemann existence theorem for stacks etc. We have tried to be as general as possible and make minimal assumptions in the definition topological stacks. Examples of topological stacks include orbifolds, graphs of groups, and complexes of groups. We also show that to any algebraic stack of finite type over C one can associate a topological stack. In particular, in all these situations, we have the machinery of homotopy theory at our disposal. Remark : Some part of this paper will be removed and be subsumed in "Covering theory for stack", see below.
"Foundations of topological stacks II."
This is planned to be the continuation of Foundation I, but currently I am not working on it. I will post some bits of it here though so I can refer to them in my other work.
Abstract: We prove that an abritrary topological stack X admits an epimorphism f: T --> X from a topological space T such that the base extension of f to any topological space is a weak equivalence. This allows one to associate a natural weak homtopy type to X. In the case where T can be chosed to be paracompact we show that to X we can associate a homotopy type. We present general examples when this paracompactness can be achieved. We also provide functorial formulation of these results.
Abstract: We show that the mapping stack Map(Y,X) of two topological stacks X and Y is again a topological stack provided Y is the quotient of a compact groupoid. In the case where Y is the quotient of a locally compact groupoid we prove that there is a map f: T --> Map(Y,X) from a topological space T such that the base extension of f to any topological space is a weak equivalence. This allows one to asscoaied a natural homtopy type to Map(Y,X).
"Fundamental groups of topological stacks with slice property." To appear in Algebraic and Geometric Topology, 39 pages, arXiv:0710.2615v1 [math.AT] (pdf).
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, Brown, Higgins, and Rhodes.
"On 2-group actions on stacks." In preparation.
Abstract: The group actions on stacks that arise naturally in examples are often 'weak' ones, meaning that the composition of group elements is not preserved strictly but only up to higher coherences. This makes the study of such actions practically impossible. In this paper we work out a machinery for the study of actions of a presheaf of weak 2-groups on a stack. Using some homotopy theory, we show that essentially one can break down such a problem into two simpler problems: a group extension problem, and a 'strict' action problem. This then applies to the case of 2-group scheme actions on algebraic stacks, Lie 2-group actions on differentiable stacks and so on.
"Covering theory for stacks." In preparation.
Abstract: We develop a general covering space theory for topological stacks. In particular, for a locally path connected semilocally 1-connected topological stack X, we establish an equivalence between the category of pi_1(X)-sets and the category of covering stacks of X.
"Notes on 2-groupoids, 2-groups, and crossed-modules." Homology, Homotopy and Applications, Vol. 9 (2007), no. 1, 75--106 (pdf). Erratum.
Abstract: This paper contains many basic results on 2-groupoids, with special emphasis on computing derived mapping groupoids between 2-groupoids. Some of the results proven here are presumably folklore (but do not appear in the literature to my knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2-groupoids introduced by Moerdijk and Svensson.
"On weak maps between 2-groups." Submitted, 48 pages, arXiv:math/0506313v3 [math.CT] (pdf).
Abstract: The main result of this paper is an explicit description of the groupoid of weak morphisms between two crossed-modules. Several approaches are considered and their compatibility is verified.
"On the structure of weighted projective general linear 2-groups." Submitted, 18 pages, arXiv:0704.1010v1 [math.AG] (pdf).
Abstract: For a given sequence of positive integers (n_0,...,n_r) we define the weighted projective general linear 2-group scheme PGL(n_0,...,n_r) as a crossed-module in the category of schemes and show that it is a model for (i.e is naturally weakly homotopy equivalent to) the (a la Breen) of the self-equivalences of the weighted projective stack of weight (n_0,...,n_r). We also give an explicit description of the structure of PGL(n_0,...,n_r). This enables us to apply the machinery developed in "On 2-group actions on stacks" (see below) to study group (scheme) actions on weighted projective stacks.
"String topology for stacks," with Kai Behrend, Gregory Ginot, Ping Xu. Preprint, 89 pages, arXiv:0712.3857v1 [math.AT] (pdf).
A small fragment of this work has appeared as "String topology for loop stacks," C. R. Math. Acad. Sci. Paris 344 (2007), no. 4, 247--252 (pdf).
Abstract: We prove that the homology groups of the loop stack of an oriented stack are equipped with a canonical loop product and loop coproduct, which makes it into a Frobenius algebra. Moreover, the shifted homology groups of the loop stack admit a BV algebra structure.
"Explicit HRS-tilting." Submitted, 31 pages, to appear in Journal on Noncommutative Geometry (pdf).
Abstract: We give an alternative, and very explicit, description of the abelian category B associated to a torsion pair (T,F) in an abelian category A. We then use this to give an explicit description of the category Ch(B) of chain complexes in B, the derived category D(B), and the DG structure on Ch(B) in terms of what we call decorated complexes .
"Generalized Fulton-MacPherson bivariant theories." In preparation.
Abstract: We introduce a slightly more general version of Fulton-MacPherson bivariant theories in which the product is only partially defined. We then show how to construct various generalized bivariant theories on topological stacks out of a given multiplicative cohomology theory for topological spaces (e.g., singular homology, K-theory, complex cobordism, etc.) This, in particular, gives rise to the construction of certain Gysin maps, characteristic classes, etc. in the setting of topological stacks.
Abstract: It is shown that the Picard stack of a weighted projective stack over an arbitrary base S is isomorphic to Z x BG_m, where BG_m stands for the classifying stack of the multiplicative group scheme over S.
Abstract: Expository notes on topological stacks based on a talk given in M.P.I.M. Oeberseminar.
Abstract: These are notes of the lectures given in Workshop on Noncomutative Geometry, Sep 11-22, 2005, I.P.M., Tehran, Iran.