Pushforwards and Gauge Transformations for Categorical Connections

On $3$-gauge transformations, $3$-curvature and $\mathbf{Gray}$-categories

In the $3$-gauge theory, a $3$-connection is given by a $1$-form $A$ valued in the Lie algebra $ \mathfrak g$, a $2$-form $B$ valued in the Lie algebra $\mathfrak h $ and a $3$-form $C$ valued in the Lie algebra $ \mathfrak l $, where $(\mathfrak g,\mathfrak h, \mathfrak l)$ constitutes a differential $ 2$-crossed module. We give the $3$-gauge transformations from a $3$-connection to another, and show the transformation formulae of the $1$-curvature $2$-form, the $2$-curvature $3$-form and the $3$-curvature $4$-form. The gauge configurations can be interpreted as smooth $\mathbf{Gray}$-functor…