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., to appear

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