Higher Idempotent Completion for Soergel Bimodules

Abstract: We present two applications of the concept of higher idempotent completion to higher categories relevant in link homology theory and higher representation theory. We show that singular Soergel bimodules can be recovered from Soergel bimodules through partial 2-categorical idempotent completions. Specializing to type A, we further assemble singular Soergel bimodules into a semistrict monoidal 2-category and identify certain quotients as semistrict monoidal 2-categories of gl_N foams. Our second main result uses 2-categorical idempotent completions to formulate a higher categorical branching rule for such foam theories, which underlie the Lee-Gornik-Rasmussen-Wu deformations of coloured gl_N link homology. In particular, we provide a fully local version of Rose-Wedrich's decomposition theorem on deformed coloured link homology.