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Iterated spans and classical topological field theories

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Abstract

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of higher-categorical local systems, and classify their fully dualizable objects. By the Cobordism Hypothesis, these give rise to framed topological quantum field theories, which are the framed versions of the classical topological quantum field theories considered in the quantization programme of Freed–Hopkins–Lurie–Teleman. Using this machinery, we also construct an \((\infty ,1)\)-category of symplectic derived algebraic stacks and Lagrangian correspondences and show that all its objects are dualizable.

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Acknowledgements

I first learned about \(\infty \)-categories of spans from conversations with Clark Barwick back in 2010. The present work was inspired by a number of discussions during my visit to the MSRI programme on algebraic topology in the spring of 2014, in particular with Hiro Tanaka and Owen Gwilliam. I also thank Oren Ben-Bassat, Damien Calaque, Theo Johnson-Freyd, Gregor Schaumann, Claudia Scheimbauer, Chris Schommer-Pries, and Peter Teichner for helpful comments.

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  1. University of Copenhagen, Copenhagen, Denmark

    Rune Haugseng

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Haugseng, R. Iterated spans and classical topological field theories. Math. Z. 289, 1427–1488 (2018). https://doi.org/10.1007/s00209-017-2005-x

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