Iterated spans and classical topological field theories

TL;DR

This paper constructs higher categories of iterated spans with additional structures, classifies their dualizable objects, and uses the Cobordism Hypothesis to produce framed topological quantum field theories, connecting to classical TQFTs.

Contribution

It introduces a framework for higher categories of iterated spans with local systems and classifies their dualizable objects, enabling the construction of framed TQFTs.

Findings

Classified fully dualizable objects in higher categories of spans.

Constructed an infinity-category of Lagrangian correspondences with all objects fully dualizable.

Established a link between these structures and classical topological quantum field theories.

Abstract

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of "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" TQFTs considered in the quantization programme of Freed-Hopkins-Lurie-Teleman. Using this machinery, we also construct an infinity-category of Lagrangian correspondences between symplectic derived algebraic stacks and show that all its objects are fully dualizable.