Higher holonomies: Comparing two constructions

We compare two different constructions of higher-dimensional parallel transport. On the one hand, there is the two-dimensional parallel transport associated with 2-connections on 2-bundles studied by Baez–Schreiber [2], Faria Martins–Picken [11] and Schreiber–Waldorf [12]. On the other hand, there are the higher holonomies associated with flat superconnections as studied by Igusa [7], Block–Smith [3] and Arias Abad–Schätz [1]. We first explain how by truncating the latter construction one obtains examples of the former. Then we prove that the two-dimensional holonomies provided by the two approaches coincide.