Generalized forms of types N = 1, 2 and higher gauge theory

In this paper, we give a compact formulation of strict higher gauge theory based on generalized (differential) forms that package fields of multiple form degrees into a single variable. We define generalized forms valued in higher algebras and higher groups and derive the corresponding Maurer--Cartan structures. This leads to uniform, gauge-theory-like expressions for higher connections, curvatures, Bianchi identities, and gauge transformations. We further construct action principles for higher Chern--Simons and higher Yang--Mills theories within the same formalism and compute the associated topological densities in the corresponding dimensions.