Cohomological twisting of 2-linearization and extended TQFT

In this paper, we describe a relation between a categorical quantization construction, called “2-linearization”, and extended topological quantum field theory (ETQFT). We then describe an extension of the 2-linearization process which incorporates cohomological twisting. The 2-linearization process assigns 2-vector spaces to (finite) groupoids, functors between them to spans of groupoids, and natural transformations to spans between these. By applying this to groupoids which represent the (discrete) moduli spaces for topological gauge theory with finite group G , the ETQFT obtained is the untwisted Dijkgraaf–Witten (DW) model associated to G . This illustrates the factorization of TQFT into “classical field theory” valued in groupoids, and “quantization functors”, which has been described by Freed, Hopkins, Lurie and Teleman. We then describe how to extend this to the full DW model, by using a generalization of the symmetric monoidal bicategory of groupoids and spans which incorporates cocycles. We give a generalization of the 2-linearization functor which acts on groupoids and spans which have associated cohomological data. We show how the 3-cocycle ω on the classifying space B G which appears in the action for the DW model induces a classical field theory valued in this bicategory.