Notes on 1- and 2-Gerbes

  title={Notes on 1- and 2-Gerbes},
  author={Lawrence S. Breen},
  journal={arXiv: Category Theory},
  • L. Breen
  • Published 10 November 2006
  • Mathematics
  • arXiv: Category Theory
The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class. 
Differential Geometry of Gerbes and Differential Forms
We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes due to W. Messing and the author [5], and give a more direct derivation of the associated
Derived 2-functors in (2-SGp)
In this paper, we construct the projective resolution of arbitrary symmetric 2-group, define the derived 2-functors in (2-SGp) and give some related properties of the derived 2-functors.
Nonabelian bundle 2-gerbes
We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called
Higher Dimensional Homology Algebra III:Projective Resolutions and Derived 2-Functors in (2-SGp)
In this paper, we will define the derived 2-functor by projective resolution of any symmetric 2-group, and give some related properties of the derived 2-functor.
Central Extensions of Gerbes
We introduce the notion of central extension of gerbes on a topo-logical space X. We show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also
Twisting by a Torsor
Twisting by a G-torsor an object endowed with an action of a group G is a classical tool. For instance one finds in the paragraph 5.3 of the book [17] the description of the “operation de torsion” in
Bicategorical homotopy fiber sequences
Small Bénabou’s bicategories and, in particular, Mac Lane’s monoidal categories, have well-understood classifying spaces, which give geometric meaning to their cells. This paper contains some
Equivariant Gromov-Witten theory of one dimensional stacks
In math.AG/0207233, Okounkov and Pandharipande gave an operator formalism for computing the equivariant Gromov-Witten theory of the projective line. This thesis extends their result to orbifold


Differential geometry of gerbes
This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of
On the correspondence between gerbes and bouquets
Duskin [ 1, 2 ] showed that gerbes over a Grothendieck topos E can be interpreted by certain internal groupoids of E which he called bouquets. He proved that for any bouquet B of E the fibred
On weak maps between 2-groups
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
Lectures on Special Lagrangian Submanifolds
These notes consist of a study of special Lagrangian submanifolds of Calabi-Yau manifolds and their moduli spaces. The particular case of three dimensions, important in string theory, allows us to
On cocycle bitorsors and gerbes over a Grothendieck topos
The aim of this paper is to give a new description of Giraud's nonabelian cohomology set H 2 ( L ), [7], defined for a band L of a Grothendieck topos E as the set of L -equivalence classes of L
Bundle gerbes
Just as C principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral
An outline of non-abelian cohomology in a topos : (I) The theory of bouquets and gerbes
© Andrée C. Ehresmann et les auteurs, 1982, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les
Homotopy Transition Cocycles
For locally homotopy trivial fibrations, one can define transition functions gfifl : Ufi \ Ufl ! H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle