  The homogeneous coordinate ring of the Grassmannian $\rm{Gr}(k,n)$ has a
well-known cluster structure. There is a categorification of this cluster
structure via a category of modules for a ring $\mathcal{A}_{k,n}$ due to
Jensen-King-Su, building on work of Geiss-Leclerc-Schr\"oer, in which cluster
variables correspond to indecomposable rigid modules. We give a combinatorial
description of modules that correspond to rank $2$ cluster variables by using
webs. This conjecturally gives the categorification of all rank $2$ cluster
variables.
