Principal Bundle Groupoids versus Vector Bundle Groupoids in Differential Geometry

Explore a mathematical lecture that delves into the relationship between principal bundle groupoids (PB-groupoids) and vector bundle groupoids (VB-groupoids). Begin with a review of how linear frames of smooth n-manifolds define principal GL(n, R)-bundles and how principal bundles with representations correspond to vector bundles. Learn about VB-groupoids and their applications in representations up to homotopy and deformation theory, before discovering a special class of frames that interact with groupoid structures. Examine how these frames connect VB-groupoids to diagrams of Lie groupoids and principal bundles through the action of a strict Lie 2-groupoid GL(l, k). Master the generalization of standard correspondence between vector bundles and principal bundles to a correspondence between VB-groupoids and PB-groupoids, concluding with examples and potential future applications of this mathematical framework.