Quantization of local systems over finite homotopy types

We explicitly define the quantization functor Sum: Famn(C) → C for C-valued representations of finite homotopy types, whose existence is suggested in [FHLT], in the case when n = 1 and C is a symmetric monoidal category with duals. Our construction is mainly based on dualizability, in line with the Cobordism Hypothesis, and does not require the isomorphism conditions assumed in [FHLT], which are instead deduced as a corollary. We obtain in fact a canonical Wirthmu¨ller-like isomorphism between certain right and left Kan extensions, thus answering an ambidexterity problem in the context of finite homotopy 1-types and categories with duals. For C = Vect the construction recovers the known Nakayama isomorphism between induced and coinduced representations of essentially finite groupoids, and we exhibit some familiar results from representation theory as consequences of the present general machinery. While our work mainly considers standard categories, our arguments have homotopical counterparts and the formalism we propose should well adapt to the (∞,n)-setting. The construction of the higher quantization functors and of the associated extended topological field theories are the subject of ongoing research.