We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of $SL_r$ webs, and is defined using the $r$-fold dimer model on the network. When $r$ equals 1, our map is Postnikov's boundary measurement used to coordinatize positroid strata. The main result is that the higher rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of $SL_r$ webs, reproving a result of Cautis-Kamnitzer-Morrison in the classical ($q=1$) setting. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity.
This is joint work with Thomas Lam and Ian Le.