Skein Categories in Non-semisimple Settings

We introduce a version of skein categories of surfaces which depends on a tensor ideal in a linear ribbon category, thereby extending the existing theory to the setting of non-semisimple TQFTs. We obtain modified notions of skein algebras of surfaces and skein modules of 3-cobordisms for non-semisimple ribbon categories. We prove that these skein categories built from ideals coincide with factorization homology, shedding new light on the similarities and differences between the semisimple and non-semisimple settings. The essential difference is the need to work with profunctors in the non-semisimple setting. Doing so produces a ``distinguished presheaf'' which plays the role of the distinguished object in skein categories in semisimple settings. As a consequence, we get a skein-theoretic description of factorization homology for a large class of balanced braided presentable categories, precisely all those which are expected to induce oriented categorified 3-TQFTs.