2024/10/26 by Adittya Chaudhuri, Chaudhuri, Adittya
Mathematics · Medicine · #22A22 #53C08 #58H05 #Advanced Topics in Algebra #Category Theory (math.CT) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.2411.00814
openalex publication_date 2024/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate an interplay between some ideas in traditional gauge theory and certain concepts in fibered categories. We accomplish this by introducing a notion of a principal Lie 2-group bundle over a Lie groupoid and studying its connection structures, gauge transformations, and parallel transport. We obtain a Lie 2-group torsor version of the one-one correspondence between fibered categories and pseudofunctors. This results in a classification of our principal 2-bundles based on their underlying fibration structures. This allows us to extend a class of our principal 2-bundles to be defined over differentiable stacks presented by the base Lie groupoids. We construct a short exact sequence of VB-groupoids, namely, the 'Atiyah sequence' associated to our principal 2-bundles. Splitting and splitting up to a natural isomorphism of our Atiyah sequence, respectively, gives us notions of 'strict connections' and 'semi-strict connections' on our principal 2-bundles. We describe such connections in terms of Lie 2-algebra valued 1-forms on the total Lie groupoids. The underlying fibration structure of our 2-bundle provides an existence criterion for strict and semi-strict connections. We study the action of the 2-group of gauge transformations on the groupoid of strict and semi-strict connections, and interestingly, we observe an extended symmetry of semi-strict connections. We demonstrate an interrelationship between `differential geometric connection-induced horizontal path lifting property in traditional principal bundles' and the `category theoretic cartesian lifting of morphisms in fibered categories' by developing a theory of connection-induced parallel transport along a particular class of Haefliger paths in the base Lie groupoid of our principle 2-bundles. Finally, we employ our results to introduce a notion of parallel transport along Haefliger paths in the setup of VB-groupoids.