  We construct a dg-enhancement of KLRW algebras that categorifies the tensor
product of a universal $\mathfrak{sl}_2$ Verma module and several integrable
irreducible modules. When the integrable modules are two-dimensional, we
construct a categorical action of the blob algebra on derived categories of
these dg-algebras which intertwines the categorical action of
$\mathfrak{sl}_2$. From the above we derive a categorification of the blob
algebra.
