Parallel transport and holonomy are well-known as notions in the tangent bundle of a Riemannian manifold. More abstractly, parallel transport and holonomy can be studied in vector bundles or principal bundles with connections. In the context of higher-categorical geometry I study analogous notions for gerbes and so-called 2-vector bundles. Here, parallel transport and holonomy are not evaluated along path but along surfaces. Completely new aspects arise, such as the fact that while every line is orientable, there exist surfaces that are not orientable, for instance the Klein Bottle. My work about higher-categorical parallel transport and holonomy concerns foundational aspects, e.g. the precise formulation of what parallel transport along a surface actually is, and the comparison of the various different versions.
Below are some articles and manuscripts of talks about this topic.
Articles associated with this topic
- Smooth Functorial Field Theories from B-Fields and D-Branes
together with Severin Bunk
J. Homotopy Relat. Struct. 16.1 (2021): 75-153
arxiv:1911.09990 - Parallel transport in principal 2-bundles
Higher Structures 2(1):57-115, 2018
arxiv:1704.08542 - A global perspective to connections on principal 2-bundles
Forum Math. 30 (2017), no. 4, 809-843
arxiv:1608.00401 - Local Theory for 2-Functors on Path 2-Groupoids
together with Urs Schreiber
J. Homotopy Relat. Struct. (2016) 1-42
arxiv:1303.4663 - Connections on non-abelian Gerbes and their Holonomy
together with Urs Schreiber
Theory Appl. Categ., Vol. 28, 2013, No. 17, pp 476-540
arxiv:0808.1923 - Smooth Functors vs. Differential Forms
together with Urs Schreiber
Homology, Homotopy Appl., 13(1), 143-203 (2011)
arxiv:0802.0663 - Parallel Transport and Functors
together with Urs Schreiber
J. Homotopy Relat. Struct. 4, 187-244 (2009)
arxiv:0705.0452
Talks associated with this topic
- Parallel Transport Functors of Principal Bundles and (non-abelian) Bundle Gerbes
Conference "Principal Bundles, Gerbes and Stacks", Physikzentrum Bad Honnef, June 2007
Notes - Parallel Transport and Functors
Conference "Categories in Geometry and mathematical Physics", Mediterranean Institute For Life Sciences, September 2007
Notes - Transport Functors and Connections on Gerbes
Seminar "Topology", University of California at Berkeley, August 2008
Notes Part 1 Notes Part 2 - Smooth Functors for higher-dimensional Parallel Transport
Workshop "Smooth Structures in Logic, Category Theory and Physics", Ottawa University, May 2009
Notes - An introduction to higher parallel transport
Seminar "Algebraic and combinatorial perspectives in the mathematical sciences", online, September 2020
Presentation Video