Invertible braided tensor categories

TL;DR

The paper proves that a finite braided tensor category is invertible in the Morita 4-category BrTens if and only if it is non-degenerate, yielding invertible 4D framed TFTs via the cobordism hypothesis and extending known modular results to non-semisimple cases. It develops a broad, general framework for dualizability and invertibility of $E_1$- and $E_2$-algebras in symmetric monoidal $(\infty,2)$-categories, culminating in precise invertibility criteria in terms of nondegeneracy, factorizability, and cofactorizability for $E_2$-algebras. The finite braided tensor category case is analyzed through centers, coends, and end structures, showing equivalences among CP-rigidity, dualizability, and invertibility, and connecting these to Drinfeld and Müger centers via canonical Hopf structures. The Picard group of BrTens and its relation to the Witt group of non-degenerate braided fusion categories is developed, with open questions about higher Morita phenomena, injectivity/surjectivity to BrTens, and the homotopy type of the Picard $(\infty)$-groupoid. Together, the results provide a non-semisimple generalization of Crane–Yetter–Kauffman-type theories and a broader paradigm for understanding invertibility in higher Morita theory.

Abstract

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the semisimple case. More generally, we characterize invertibility for E_1- and E_2-algebras in an arbitrary symmetric monoidal oo-category, and we conjecture a similar characterization of invertible E_n-algebras for any n. Finally, we propose the Picard group of BrTens as a generalization of the Witt group of non-degenerate braided fusion categories, and pose a number of open questions about it.

Figures (4)

• Figure 1: Multiplication $m$, coproduct $\Delta$, counit $\epsilon$ and antipode $S$ on $\mathcal{F}$.
• Figure 2: Hopf self-pairing $\omega$ and the isomorphism $\tau_V$.
• Figure 3: Drinfeld map $\operatorname{Dr}\colon\mathcal{F}\rightarrow \mathcal{E}$.
• Figure 4: The action map $\mathcal{F}\otimes \mathrm{triv}_r(y)\rightarrow \mathrm{triv}_r(y)$ and the corresponding map $\mathcal{F}\rightarrow y\otimes y^\vee$.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: ShimizuND
• Theorem 1.6
• Remark 1.7
• Conjecture 1.8
• Conjecture 1.9
• Definition 2.1