In this paper we establish a one-to-one correspondence between S 1-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with presently working as a postdoc at the University of Nottingham, UK 1 group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 1, satisfying a smoothness condition, where a homotopy between maps from [0,1] 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids. 1

(0.1) The goal of this paper is to develop a certain mathematical framework underlying the S-duality conjecture of Vafa and Witten [VW]. Let us recall the formulation. Let S be a smooth projective surface over C and G a semisimple algebraic group. Denote by MG(S,n) the moduli space of semistable principal G-bundles on

Recent developments in higher-dimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2-categories, and a center construction for monoidal 2-categories which gives a braided monoidal 2-category. I give generalized center constructions for braided and sylleptic monoidal 2-categories which give sylleptic and symmetric monoidal 2-categories respectively, and I correct some errors in the original center construction for monoidal 2-categories. 1 Introduction The initial motivation for the study of braided monoidal categories was twofold: from homotopy theory, where braided monoidal categories of a particular kind arise as algebraic 3-types of arc-connected, simply connected spaces, and from higher-dimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16]. These motivations have subsequently been brought together by the definition of tricategori...

The purpose of this paper is to develop some additional techniques for the weak n-categories defined by Tamsamani in [27] (which he calls n-nerves). The goal is to be able to define the internal Hom(A, B) for two n-nerves A and B, which should itself be an n-nerve. This in turn is for defining the n + 1-nerve nCAT of all n-nerves conjectured in

this article, but beware of misprints. A more thorough treatment is given in May, [18]. We will use (standard) notation from [18] wherever possible. The way found initially around the restriction that K had to be reduced in the above loop construction was to take a maximal tree in K and to contract it to a point. In 1984, a groupoid version of the loop group construction was given by Dwyer and Kan, [12]. (Unfortunately the published paper has many misprints and the cleaned-up version that we will use was prepared by my student Phil Ehlers as part of his master's dissertation, [13]. Alternatives have been proposed by Joyal and Tierney, and by Moerdijk and Svensson. They end up with simplicial objects in the category of groupoids, whilst the Dwyer - Kan version gives a simplicially enriched groupoid, i.e. a groupoid all of whose Hom-objects are simplicial sets. A simplicially enriched groupoid is also a simplicial groupoid (simplicial object in the category of groupoids), but is one whose object of objects is a constant simplicial set.) Let SS denote the category of simplicial sets and SGpds that of simplicially enriched groupoids or as we will often call them, simply, simplicial groupoids. The loop groupoid functor is a functor

In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for n-categories of the well-known stabilization theorems in homotopy theory. To explain the statement, recall that Baez-Dolan introduce the notion of k-uply monoidal n-category which is an n + k-category having only one i-morphism for all i < k. This includes the notions previously defined and examined by many authors, of monoidal (resp. braided monoidal, symmetric monoidal) category (resp. 2-category) and so forth, as is explained in [2] [4]. See the bibliographies of those preprints as well as that of the the recent preprint [9] for many references concerning these types of objects. In the case where the n-category in question is an n-groupoid, this notion is—except for truncation at n—the same thing as the notion of k-fold iterated loop space, or “Ek-space ” which appears in Dunn [10] (see also some anterior references from there). The fully stabilized notion of k-uply monoidal n-categories for k ≫ n is what Grothendieck calls Picard n-categories in [12]. The stabilization hypothesis [2] states that for n + 2 ≤ k ≤ k ′ , the k-uply monoidal

I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2-transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2-space is the 2-space corresponding to a given space and the 2-group is the automorphism 2-group of a given group, then this 2-category is equivalent to the 2-category of gerbes over that space under that group (being described by the same cohomological data).

(0.1) The goal of this paper and the next one [KV2] is to relate three subjects of recent interest: (A) The theory of sheaves of chiral differential operators (CDO), see [GMS1-2]. A sheaf of CDO on a complex manifold X is a sheaf of graded vertex algebras with certain conditions on the graded components. As shown in loc. cit., locally on X such an object always exists and is unique up to an isomorphism but the isomorphism not being canonical, the global situation is similar to the behavior of spinor bundles on a Riemannian manifold. This is expressed by saying that sheaves of CDO form a gerbe CDOX. A global object exists if and only if the characteristic class (0.1.1) ch2(X) = 1 2 c21 (X) − c2(X) vanishes. Manifolds with this property are known as MU〈8〉-manifolds in homotopy theory. (B) The theory of the group GL(∞) developed by Sato and others [PS], in particular,

One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop

Consider a ring K, a topological space X and a sheaf A on X of K[[�]]-algebras. Assuming A �-adically complete and without �-torsion, we first show how to deduce a coherency theorem for complexes of A-modules from the corresponding property for complexes of A /�A-modules. We apply this result to prove that, under a natural properness condition, the convolution of two coherent kernels over deformation quantization algebroids on complex Poisson manifolds is coherent. We also construct the dualizing complexes for such algebroids