  We construct a modular functor which takes its values in the monoidal
bicategory of finite categories, left exact functors and natural
transformations. The modular functor is defined on bordisms that are 2-framed.
Accordingly we do not need to require that the finite categories appearing in
our construction are semisimple, nor that the finite tensor categories that are
assigned to two-dimensional strata are endowed with a pivotal structure. Our
prescription can be understood as a state-sum construction. The state-sum
variables are assigned to one-dimensional strata and take values in bimodule
categories over finite tensor categories, whereby we also account for the
presence of boundaries and defects. Our construction allows us to explicitly
compute functors associated to surfaces and representations of mapping class
groups acting on them.
