Iterated spans and classical topological field theories
type: Item
instance of: scholarly article
title: Iterated spans and classical topological field theories
zbMATH Open document ID: 1400.18006
author: Rune Haugseng
published in: Mathematische Zeitschrift
publication date: 2018-08-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1409.0837
review text: 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.
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 81R10
zbMATH DE Number: 6928370
MaRDI profile type: Publication
OpenAlex ID: W1636088426
arXiv ID: 1409.0837
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DOI: 10.1007/S00209-017-2005-X
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description: scientific article