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We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self‐contained review on simplicial sets as models of (∞, 1)‐categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Ševera, that maps higher groupoids to    L  ∞   ‐algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six‐dimensional superconformal field theories via a Penrose–Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non‐Abelian self‐dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.

Higher groupoid bundles, higher spaces, and self‐dual tensor field equations