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In the study of 3d N=4 gauge theories, I became interested in the category of matrix factorizations, and its definition for infinite dimensional spaces. In this talk, I will talk about the definition of the category of matrix factorizations, its properties, Koszul dualities, and the places it shows up in 3d N=4 gauge theories. I will explain how in this context, one can relate this category to representations of quantum groups, and match a derived equivalence up to fusion. Koszul duality will relate this category to coherent sheaves on the Higgs branch. These considerations are inspired while working on the 3d N=4 mirror symmetry project with A. Ballin, T. Creutzig and T. Dimofte.