Category objects and groupoid gerbes

Abstract

We introduce the notions of category objects and groupoid objects in a category C with finite fibre products. A groupoid object is a category object with an inversion structuremorphismwhich sometimes is unique. In this case, a groupoid is a category object with an axiom, otherwise it is a category with additional structure. For this, we begin with n-level pseudosimplicial objects in a category in order to formulate the notion of a category object. These have been used already for a description of local triviality of bundles and the construction of bundles from local data or from descent data. Then, we go further with simplicial objects, geometric realization, the nerve of a category, and the final step in the detour away fromgerbes to the definition of algebraic K-theory. All this illustrates the vast influence of these general concepts in mathematics for which there are more and more applications to physics. Groupoid gerbes are line bundles over the morphism space G(1) in a groupoid G(*), which in case G = G(1) is a group, correspond to central extensions. In the case of bundle gerbes, these groupoid gerbes are line bundles L over the fibre product Y × BY of a space Y over B together with a multiplication which uses the fibre product structure. Moreover, for bundle gerbes, there is a line bundle isomorphism on Y × BY × BY with an associativity condition on Y ×B Y × BY × BY, and for groupoid gerbes, there is a line bundle isomorphism on G(1)×G(0)G(1) with an associativity condition on G(1)×G(0)G(1)×G(0)G(1). The line bundle on G(1) usually is taken to be symmetric with respect to the inverse mapping of the groupoid. © Springer-Verlag Berlin Heidelberg 2008.