  We provide a visual and intuitive introduction to effectively calculating in
2-groups along with explicit examples coming from non-abelian 1- and 2-form
gauge theory. In particular, we utilize string diagrams, tools similar to
tensor networks, to compute the parallel transport along a surface using
approximations on a lattice. Although this work is mainly intended as
expository, we prove a convergence theorem for the surface transport in the
continuum limit. Locality is used to define infinitesimal parallel transport
and two- dimensional algebra is used to derive finite versions along arbitrary
surfaces with sufficient orientation data. The correct surface ordering is
dictated by two-dimensional algebra and leads to an interesting diagrammatic
picture for gauge fields interacting with particles and strings on a lattice.
The surface ordering is inherently complicated, but we prove a simplification
theorem confirming earlier results of Schreiber and Waldorf. Assuming little
background, we present a simple way to understand some abstract concepts of
higher category theory. In doing so, we review all the necessary categorical
concepts from the tensor network point of view as well as many aspects of
higher gauge theory.
