# Publications

**Fundamental groups of algebraic stacks**[pdf] [arxiv] [journal]*J. Inst. Math. Jussieu,*3 (2004), no. 1, 69-103.**Abstract.**We develop a Grothendieck-Galois theory of finite etale covers for algebraic stacks. Read on**Abstract.**We develop a Grothendieck-Galois theory of finite etale covers for algebraic stacks. We show that the corresponding Galois groups are equipped with an additional structure, namely a homomorphism from the inertia group. We show that these homomorphisms govern the stacky structure of the covering stacks. We use these homomorphisms to: 1) give a simple classification of algebraic stacks which are quotient stacks of finite group actions, 2) a formula for the fundamental group of the coarse moduli space. This is part of the author's M.I.T thesis. Hide abstract**Uniformization of Deligne-Mumford analytic curves**[with K. Behrend] [pdf] [arxiv] [journal]*J. Reine Angew. Math.,*599 (2006), 111-153.**Abstract.**We give a complete classification of smooth analytic Deligne-Mumford curves. Read on**Abstract.**We give a complete classification of smooth analytic Deligne-Mumford curves. We give a canonical presentation of each such curve as a quotient stack and compute its fundamental group. A main technical input is the theory of crossed-modules and butterflies. We also discuss connections with the theory of F-groups and Bass-Serre theory of graphs of groups. Hide abstract**Notes on 2-groupoids, 2-groups, and crossed-modules**[pdf] [arxiv] [journal]*Homology, Homotopy and Applications,*9 (2007), no. 1, 75-106.**Abstract.**This paper studies the derived mapping space between two 2-groupoids. Read on**Abstract.**This paper concerns the derived mapping space between two 2-groupoids. We clarify the relation between the derived mapping space and the 2-groupoid of weak functors. We also study the effect of strictification of weak 2-groupoids on the dervied maping spaces. We also consider the pointed case and discuss its relation to weak morphisms of crossed-modules.

*Remark.*The pdf version is more up-to-date than the published version. An error in Definition 8.4 has been fixed. Hide abstract**String topology for loop stacks**[with K. Behrend, G. Ginot and P. Xu] [pdf]*C. R. Math. Acad. Sci. Paris,*344 (2007), no. 4, 247-252.**Abstract.**We extend Chas-Sullivan's theory to oriented topological stacks. Read on**Abstract.**We extend Chas-Sullivan's theory to oriented stacks. We prove that the homology groups of the loop stack of an oriented topological stack are equipped with a canonical loop product and loop coproduct, which makes it into a Frobenius algebra. Moreover, we show that the shifted homology groups of the loop stack admit a Batalin-Vilkovisky algebra structure. Hide abstract**Lectures on derived and triangulated categories**[pdf] [arxiv]In

*Invitation to Noncommutative Geometry,*M. Khalkhali and M. Marcolli (editors), World Scientific, 2007.**Abstract.**Lectures at*Workshop on Noncomutative Geometry*, Sep 11-22, 2005, I.P.M., Tehran. Topics discussed: additive and abelian categories, abelian categories of sheaves, reconstruction theorems, derived categories, triangulated categories, t-structures and tilting. Hide abstract**Fundamental groups of topological stacks with slice property**[pdf] [arxiv] [journal]*Algebraic and Geometric Topology,*8 (2008), 1333-1370.**Abstract.**We give a formula for the fundamental group of the coarse moduli of a topological stack. Read on**Abstract.**We give a formula for the fundamental group of the coarse moduli of a topological stack. As an application, we find simple general formulas for the fundamental group of the coarse quotient of a compact Lie group action on a topological space in terms of the fixed point data. We recover, and vastly generalize, results of Armstrong, Bass, Brown, Higgins, and Rhodes. The notion of slice property for a stack, which plays an essential role in the paper, may also be of independent interest. It is a stacky formulation of the slice property of compact Lie group actions. Hide abstract**Explicit HRS-tilting**[pdf] [arxiv]*Journal of Noncommutative Geometry,*3 (2009), no. 2, 223-259.**Abstract.**We given an explicit description of the tilted category B associated to a torsion pair. Read on**Abstract.**We given an explicit description of the tilted category B associated to a torsion pair. We also give an explicit description for the category Ch(B) of chain complexes in B, the derived category D(B), and the DG structure on Ch(B). As a consequence, we find new proofs of certain results of Happel-Reiten-Smalo. The main ingredient is the category of decorated complexes. Hide abstract**Butterflies I: morphisms of 2-group stacks**[with E. Aldrovandi] [pdf] [arxiv] [journal]*Advances in Mathematics,*221 (2009), no. 3, 687-773.**Abstract.**In this paper we use the notion of butterfly to give an efficient description of... Read on**Abstract.**In this paper we use the notion of butterfly to give an efficient description of morphisms and 2-morphisms between 2-group stacks as well as the weak morphisms between sheaves of crossed-modules over a Grothendieck site. We prove the 7-term long exact sequence associated to a non-abelian “exact triangle” determined by a morphism of 2-group stacks. We also study the cases of symmetric and Picard stacks. Deligne’s results in [SGA 4, Exp. XVIII] relating the derived category of complexes of abelian sheaves to the category of Picard stacks follow immediately, and get strengthened, using our theory. We also obtain non-abelian versions of Deligne’s results. Hide abstract**Mapping stacks of topological stacks**[pdf] [arxiv] [journal]*J. Reine Angew. Math.,*646 (2010), 117-133.**Abstract.**We show that the mapping stack Map(Y,X) is a topological stack if Y is compact. Read on**Abstract.**We show that the mapping stack Map(Y,X) is a topological stack if Y is compact. I.e., if Y admits a compact groupoid presentation. If Y admits a locally compact 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 application, we describe the loop stack of the classifying stack BG of a topological group G in terms of twisted loop groups of G.

*Correction.*In Proposition 5.2.(ii) the phrase 'closed embedding' should be removed. See the paper ''Fibrations of topological stacks'' (below) for the correct statement. Hide abstract**Butterflies II: torsors for 2-group stacks**[with E. Aldrovandi] [pdf] [arxiv] [journal]*Advances in Mathematics,*225 (2010), no. 2, 922-976.**Abstract.**We study torsors over 2-groups and their morphisms. Read on**Abstract.**We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We re-interpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a crossed module. Our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the ``fraction'' (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at the cocycle level. In addition, we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions. Hide abstract**Group cohomology with coefficients in a crossed-module**[pdf] [arxiv] [journal]*J. Inst. Math. Jussieu,*10 (2011), no. 2, 359-404.**Abstract.**We compare three approaches to cohomology with coefficients in a crossed-module. Read on**Abstract.**We compare three approaches to 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. Hide abstract**String topology for stacks**[with K. Behrend, G. Ginot and P. Xu] [pdf] [arxiv]*Astérisque*343 (2012).**Abstract.**We establish the general machinery of string topology for oriented differentiable stacks. Read on**Abstract.**We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on equal footing free loops in stacks and hidden loops. We construct a bivariant (in the sense of Fulton and MacPherson) theory for topological stacks: it gives us a flexible theory of Gysin maps which are automatically compatible with pullback, pushforward and products. Further we prove an excess formula in this context. We introduce oriented stacks, generalizing oriented manifolds, which are stacks on which we can do string topology. We prove that the homology of the free loop stack of an oriented stack and the homology of hidden loops (sometimes called ghost loops) are a Frobenius algebra which are related by a natural morphism of Frobenius algebras. We also prove that the homology of free loop stack has a natural structure of BV-algebra, which together with the Frobenius structure fits into an homological conformal field theories with closed positive boundaries. Using our general machinery, we construct an intersection pairing for (non necessarily compact) almost complex orbifolds which is in the same relation to the intersection pairing for manifolds as Chen-Ruan orbifold cup-product is to ordinary cup-product of manifolds. We show that the hidden product of almost complex is isomorphic to the orbifold intersection pairing twisted by a canonical class. Finally we gave some examples including the case of the classifying stacks [*/G] of a compact Lie group. Hide abstract**Homotopy types of topological stacks**[pdf] [arxiv] [journal]*Advances in Mathematics,*230 (2012), 2014-2047.**Abstract.**We define the notion of classifying space of a topological stack Read on**Abstract.**We define the notion of classifying space of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain paracompactness condition on \X, we show that X is actually well-defined up to homotopy equivalence. These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces. We prove similar results for (small) diagrams of topological stacks. Hide abstract**Integrating morphisms of Lie 2-algebras**[pdf] [arxiv] [journal]*Compositio Mathematicae,*149 (2013), issue 02, 264-294.**Abstract.**We show how to integrate a morphism of 2-term dglas to a weak map of Lie 2-groups. Read on**Abstract.**We show how to integrate a morphism of 2-term dglas to a weak map of Lie 2-groups. To do so we develop a theory of butterflies of 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. Hide abstract**Fibrations of topological stacks**[pdf] [arxiv] [journal]*Advances in Mathematics,*252 (2014), 612–640.**Abstract.**In this note we study fibrations of topological stacks and establish their main properties. Read on**Abstract.**In this note we study fibrations of topological stacks and establish their main properties, generalizing the corresponding theory for topological spaces. We establish the fiber homotopy exact sequence for a fibration. We also deduce van Kampen's theorem for fundamental groups of topological stacks as an immediate corollary of our results. We prove various criteria for a morphism of topological stack to be a fibration. We then use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration, generalizing the standard result for topological spaces, and use this to construct homotopy fiber of a morphism of topological stacks. We use these results to construct Leray-Serre and Eilenberg-Moore spectral sequences for fibrations of topological stacks. Hide abstract

# Preprints

**Foundations of topological stacks I**[pdf] [arxiv]**Abstract.**This is the first in a series of papers devoted to foundations of topological stacks. Read on**Abstract.**This is the first in a series of papers devoted to foundations of topological stacks. In this paper we discuss: the general formalism of topological stacks, Morita equivalence of groupoids, homotopy, homotopy groups of stacks, Deligne-Mumford topological stacks, and covering theory for stacks. We also discuss certain general classes of topological stacks: graphs-of-groups, orbifolds, topological stacks associated to Artin stacks and gerbes.*Remark.*This paper is under substantial revision. Hide abstract**On weak maps between 2-groups**[pdf] [arxiv]**Abstract.**We introduce the notion of a butterfly between crossed-modules. Read on**Abstract.**We introduce the notion of a butterfly between crossed-modules. Butterflies provide an efficient cocycle-free way of treating weak morphisms between crossed-modules. With morphisms being butterflies, crossed-modules form a bicategory which is biequivalent to the bicategory of connected pointed 2-types and also to that of 2-groups and weak morphisms. We also discuss braided and symmetric butterflies.*Remark.*This paper is under substantial revision. Hide abstract**Group actions on algebraic stacks via butterflies**[pdf] [arxiv]**Abstract.**We give an explicit description of the structure of PGL(n_1,...,n_k). Read on**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}.*Remark.*This paper is a revised version of the paper "Automorphism 2-group of a weighted projective stack". Hide abstract**Group actions on stacks and applications to equivariant string topology for stacks**[with G. Ginot] [pdf] [arxiv]**Abstract.**We construct string operations on the S^1-equivariant homology of the loop stack L(X). Read on**Abstract.**We construct string operations on the S^1-equivariant homology of the loop stack L(X) of an oriented differentiable stack X. We show that H^{S^1}_{*+dim(X) -2}(L(X)) is a graded Lie algebra. In the particular case where X is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on. Hide abstract**Singular chains on topological stacks**[with T. Coyne] [pdf] [arxiv]**Abstract.**We extend the functor Sing of singular chains to the category of topological stacks. Read on**Abstract.**We extend the functor Sing of singular chains to the category of topological stacks and establish its main properties. We prove that Sing respects weak equivalences and takes a morphism of topological stacks that is both a Serre and a Reedy fibration to a Kan fibration of simplicial sets. When restricted to the category of topological spaces Sing coincides with the usual singular functor.*Remark.*A missing condition in Lemma 9.3 of the arXiv version is added in the pdf. Hide abstract**Moduli of non-commutative polarized schemes**[with K. Behrend] [pdf] [arxiv]**Abstract.**We construct, using geometric invariant theory, a quasi-projective Deligne-Mumford Read on**Abstract.**We construct, using geometric invariant theory, a quasi-projective Deligne-Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A-infinity-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a version of Hilbert stability for non-commutative projective schemes. Hide abstract

# Miscellaneous

**Picard stack of a weighted projective stack**[pdf]**Abstract.**We show that the Picard stack of a weighted projective stack is isomorphic to Z x BG_m. Read on**Abstract.**We show that the Picard stack of a weighted projective stack is isomorphic to Z x BG_m. Everything is over an arbitrary base scheme S and BG_m stands for the classifying stack of the multiplicative group scheme over S. Hide abstract**A quick introduction to fibered categories and topological stacks**[pdf]**Abstract.**This is a quick introduction to fibered categories and topological stacks. Read on**Abstract.**This is a quick introduction to fibered categories and topological stacks. It is a slightly revised version of a few sections of [BGNX] which I thought might be useful to beginners (who may not necessarily be interested in [BGNX] itself). Hide abstract**What is a topological stack**[pdf]**Abstract.**Expository notes on topological stacks based on a talk given in M.P.I.M. Oberseminar.