This paper describes a relationship between essentially finite groupoids and two-vector spaces. In particular, we show to construct two-vector spaces of Vectvalued presheaves on such groupoids. We define two-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation-a weak functorfrom Span(FinGpd) (the bicategory of essentially finite groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.

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 2linearization functor which acts on groupoids and spans which have associated cohomological data. We show how the 3-cocycle ω on the classifying space BG which appears in the action for the DW model induces a classical field theory valued in this bicategory.

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We give an introductory account of Khovanov's categorification of the Heisenberg algebra, and construct a combinatorial model for it in a 2-category of spans of groupoids. We also treat a categorification of U (sl n ) in a similar way. These give rise to standard representations on vector spaces through a linearization process.