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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 $\mathbf{Gray}$-functors between two $\mathbf{Gray}$ $3$-groupoids: the path $3$-groupoid $\mathcal{P}_3(X)$ and the $3$-gauge group $ \mathcal{G}^{\mathscr L}$ associated to the $ 2$-crossed module $\mathscr L$, whose differential is $(\mathfrak g,\mathfrak h, \mathfrak l)$. The derivatives of $\mathbf{Gray}$-functors are $3$-connections, and the derivatives of lax-natural transformations between two such $\mathbf{Gray}$-functors are $3$-gauge transformations. We give the $3$-dimensional holonomy, the lattice version of the $3$-curvature, whose derivative gives the $3$-curvature $4$-form. The covariance of $3$-curvatures easily follows from this construction. This $\mathbf{ Gray}$-categorical construction explains why $3$-gauge transformations and $3$-curvatures have the given forms. The interchanging $3$-arrows are responsible for the appearance of terms concerning the Peiffer commutator $\{,\}$.