Iterated spans and classical topological 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.

Accepted version, plus corrections to Remarks 10.5 and 10.7. 64 pages

References in corpus (7)

• Enriched $\infty$-categories via non-symmetric $\infty$-operads
• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
• The higher Morita category of $E_n$-algebras
• Rectification of enriched infinity-categories
• Quantization via Linear homotopy types
• Bimodules and natural transformations for enriched $\infty$-categories
• The stack of higher internal categories and stacks of iterated spans

Cited by in corpus (38)

• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
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• Modules over algebraic cobordism
• Shifted cotangent stacks are shifted symplectic
• Shifted Coisotropic Correspondences
• Ambidexterity and the universality of finite spans
• Perversely categorified Lagrangian correspondences
• Symplectic implosion and the Grothendieck-Springer resolution
• Higher Semiadditive Algebraic K-Theory and Redshift
• A lax monoidal Topological Quantum Field Theory for representation varieties
• Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf-Witten theory
• Motivic theory of representation varieties via Topological Quantum Field Theories
• Bimodules and natural transformations for enriched $\infty$-categories
• Nakayama categories and groupoid quantization
• Relative critical loci and quiver moduli
• A bivariant Yoneda lemma and $(\infty ,2)$-categories of correspondences
• 2-Segal objects and the Waldhausen construction
• Derived stacks in symplectic geometry
• Shifted symplectic reduction of derived critical loci
• Simplicial spaces, lax algebras and the 2-Segal condition
• Open Systems in Classical Mechanics
• Comparison of Waldhausen constructions
• Frobenius objects in the category of spans
• The universal Hall bialgebra of a double 2-Segal space
• Frobenius and commutative pseudomonoids in the bicategory of spans
• Characters and transfer maps via categorified traces
• Localizing infinity-categories with hypercovers
• Constructing Span Categories From Categories Without Pullbacks
• Higher categories of push-pull spans, I: Construction and applications
• The Disc-structure space
• Universality of Barwick's unfurling construction
• Coherence for indexed symmetric monoidal categories
• Higher Lawvere theories
• Semisimple Field Theories Detect Stable Diffeomorphism
• A pasting theorem for iterated Segal spaces
• Higher Segal spaces and Lax ${\mathbb{A}}_{\infty }$-algebras
• A geometrical description of untwisted 3d Dijkgraaf-Witten TQFT with defects