Stacks and gerbes

Abstract

A basic structure in mathematics for the study of a space X is to give to each open set an object A(U) in a category C together with restriction morphisms rV,U: A(U) → A(V) for V ⊂ U satisfying rU,U is the identity and the composition property rW,U = rW,VrV,U for W ⊂ V ⊂ U. Such a structure is called a presheaf with values in C. Again, there is a gluing condition which is realized by a universal construction. To formulate this, we use adjoint functors which are introduced in Sect. 2. An example of such a functor is the sheaf associated to a presheaf. In this chapter, we consider another approach to gerbes by the more general concept of stack. A stack over a space X is a configuration of categories over X associated to open sets of X with various gluing properties. Instead of gluing trivial bundles to obtain a general bundle, we consider categories FU indexed by the open subsets of a space X together with restriction functors rV,U,: FU → FV for V ⊂U. Normally, we expect to have the composition property rW,U = rW,V rV,U for W ⊂ V ⊂ U of a presheaf, but now we can only assume that a third element of structure is given, namely an isomorphism between the functors rW,U and rW,V rV,U. When these data satisfy a suitable coherence relation, this triple of data is called a category over X. This concept was introduced by Grothendieck with the term fibred category over X. It is a generalization of the presheaf of categories where the key feature is that transitivity of restriction is just defined up to isomorphism. Although the entire collection of categories, denoted by F(U) or FU, over open sets do not form a presheaf, but only a presheaf up to an isomorphism in the transitivity relation, it is possible to extract from the category over X various presheaves on X. The condition that these presheaves are sheaves of sets is the first step in formulating the definition of a stack. We have seen that descent conditions play a basic role in many parts of bundle theory, and we formulate descent data concepts in a category over a space. The condition that a category over a space be a stack is formulated in terms of descent data in a category being realized as objects in the category. © Springer-Verlag Berlin Heidelberg 2008.