Iterated spans and classical topological field theories
The first objective in this paper is to construct the symmetric monoidal \((\infty,n)\)-categories of iterated spans in an \((\infty,1)\)-category (or \(\infty\)-category) \(\mathcal{C}\) with finite limit, and to classify the fully dualizable objects in these \((\infty,n)\)-categories, which, via the Cobordism Hypothesis, gives rise to framed topological quatum field theories as a simple model for classical topological field theories. The second objective in this paper is to construct variants of these \((\infty,n)\)-categories with spans being provided with local systems, and to classify their fully dualizable objects, which, via the Cobordism Hypothesis, gives rise to topological quatum field theories as classical topological field theories proposed in the quantization program of [\textit{D. S. Freed} et al., CRM Proceedings and Lecture Notes 50, 367--403 (2010; Zbl 1232.57022)]. The third objective in this paper is to apply this machinery within the context of shifted symplectic structures on derived algebraic stacks [\textit{T. Pantev} et al., Publ. Math., Inst. Hautes Étud. Sci. 117, 271--328 (2013; Zbl 1328.14027)] to construct \((\infty,1)\)-categories of \(n\)-shifted symplectic derived stacks with morphisms given by Lagrangian correspondences, in which all objects are shown to be dualizable, determining framed \(1\)-dimensional topological quantum field theories.
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