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We construct a modular functor which takes its values in the monoidalbicategory of finite categories, left exact functors and naturaltransformations. The modular functor is defined on bordisms that are 2-framed.Accordingly we do not need to require that the finite categories appearing inour construction are semisimple, nor that the finite tensor categories that areassigned to two-dimensional strata are endowed with a pivotal structure. Ourprescription can be understood as a state-sum construction. The state-sumvariables are assigned to one-dimensional strata and take values in bimodulecategories over finite tensor categories, whereby we also account for thepresence of boundaries and defects. Our construction allows us to explicitlycompute functors associated to surfaces and representations of mapping classgroups acting on them.

A modular functor from state sums for finite tensor categories and their bimodules