Iterated spans and classical topological field theories

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 TQFTs considered in the quantization programme of Freed-Hopkins-Lurie-Teleman.Using this machinery, we also construct an (∞, 1)-category of symplectic derived algebraic stacks and Lagrangian correspondences and show that all its objects are dualizable.

Equivariant higher Hochschild homology and topological field theories

“…Example 3.17 (Example for a target category: Cospans in E ∞ -algebras). For the main constructions of the present article we will be interested in the following target: Given an (∞, 1)-category C with finite colimits, we obtain by dualizing the results of [Hau17] an (∞, 1)-category Cospan(C) of cospans in C. When presented as a Segal space this (∞, 1)-category can be explicitly described in degree k ≥ 0 as follows: First denote by Σ k the partially ordered set of pairs (i, j) such that 0 ≤ i ≤ j ≤ k, where (i, j) ≤ (i , j ) if i ≤ i and j ≤ j [Hau17, Definition 5.1]. In fact, Σ • yields a cosimplicial object in categories.…”

“…By [Hau17,Corollary 8.5], Cospan(C) is a complete Segal space, the simplical structure is induced by the cosimplical structure of Σ • . Using the coproduct of C we can turn Cospan(C) into a symmetric monoidal (∞, 1)-category.…”

“…Now choose as the target (∞, n)-category the category of iterated cospans in E ∞ -Alg. By [Hau17,Theorem 13.3] every object in this category is fully dualizable.…”

Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

“…We require a decorated version of Span(Grpd), in which objects and morphisms carry compatible local systems. A general construction in the context of (∞, n)-categories has been developed in [19]. In the bicategorical setting there are explicit constructions [39], [38].…”

“…Their quantization can then be approached by postcomposing with a suitable functor to a sufficiently linear target higher category [13]. In the (∞, n)-categorical setting, such linearization functors were constructed by Haugseng [19] while in the present bicategorical setting, following earlier work of Morton [32], [33] in the case without local systems, Schweigert-Woike [39], [38] constructed a symmetric monoidal pseudofunctor…”