In higher gauge theory also gauge transformations themseves may be different but still equivalent: if there is a gauge-of-gauge transformation$\rho \colon \lambda_1 \to \lambda_2$ between them.

These higher order gauge transformations are maybe best known in the physics literature in terms of their infinitesimal approximation, the BRST complex: here the gauge transformations correspond to ghost fields and the gauge-of-gauge transformations to ghost-of-ghost fields.

Examples

A basic example of a gauge field that has higher order gauge transformations is the B-field. But also magnetic current, if described properly, exhibits higher gauge transformations, see at Dirac charge quantization.