There are several distinct frameworks within which higher gauge theories have been developed.
Alvarez et al. [1] extend the notion of integrability to higher dimensions in the context of geometric field theories.
Several works[2] of John Baez, Urs Schreiber and coauthors have developed higher gauge theories heavily based on category theory.
An alternative approach,[4] motivated by the goal of constructing geometry over spaces of paths and higher-dimensional objects, has been developed by Saikat Chatterjee, Amitabha Lahiri, and Ambar N. Sengupta.
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