Tensor product categorifications, Verma modules and the blob 2-category

We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This action intertwines with the categorical action of sl2. From the above we derive a categorification of the blob algebra.