Standard A-lattices, rigid C* tensor categories, and (bi)modules

In this article, we construct a 2 -shaded rigid C* multitensor category with canonical unitary dual functor directly from a standard A -lattice. We use the notions of traceless Markov towers and lattices to define the notion of module and bimodule over standard A-lattice(s), and we explicitly construct the associated module category and bimodule category over the corresponding 2 -shaded rigid C* multitensor category. As an example, we compute the modules and bimodules for Temperley-Lieb-Jones standard A -lattices in terms of traceless Markov towers and lattices. Translating into the unitary 2 -category of bigraded Hilbert spaces, we recover De Commer-Yamashita's classification of TL# module categories in terms of edge weighted graphs, and a classification of TL# bimodule categories in terms of biunitary connections on square -partite weighted graphs. As an application, we show that every (infinite depth) subfactor planar algebra embeds into the bipartite graph planar algebra of its principal graph.