On dualizability of braided tensor categories

TL;DR

<3-5 sentence high-level summary>This work identifies cp-rigid subcategories of the higher Morita theories of tensor and braided tensor categories that are $2$- and $3$-dualizable, respectively, and shows braided fusion categories are fully $4$-dualizable in characteristic zero. By combining these dualizability results with the cobordism hypothesis, it yields local framed topological field theories in dimensions $2$, $3$, and $4$, arising from tensor, braided tensor, and braided fusion data. The methods extend known finite/semi-simple dualizability to presentable, non-semisimple contexts and connect to quantum character theories, skein modules, and Crane--Yetter/Witten--Reshetikhin--Turaev-type invariants. This provides a unified framework for constructing higher TFTs from cp-rigid tensorial and braided structures and deepens the link between categorified Morita theory and quantum topology.

Abstract

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produce a framed 3-dimensional local TFT attached to the category of representations of a quantum group at any value of $q$.

Figures (4)

• Figure 1: Braided tensor categories as locally constant factorization algebras on $\mathbb{R}^2$. (A) depicts a basic open set; its inclusion onto $\mathbb{R}^2$ is a retract, so it is assigned the category $\mathcal{A}$ canonically. (B) depicts an embedding $\mathbb{D}^2\sqcup \mathbb{D}^2\hookrightarrow\mathbb{D}^2$; this induces the product functor $T:\mathcal{A}\boxtimes\mathcal{A}\to\mathcal{A}$. (C) depicts an isotopy (with this choice of representatives, it an identity) between two composite disk embeddings $\mathbb{D}\sqcup\mathbb{D}\sqcup\mathbb{D}\hookrightarrow\mathbb{D}$; this induces the associator natural isomorphism $\alpha$ on $\mathcal{A}$. (D) depicts an isotopy between two disk inclusions; this induces the braiding isomorphism $\sigma$ on $\mathcal{A}$.
• Figure 2: Central algebras as locally constant factorization algebras on $(\mathbb{R}\subset \mathbb{R}^2)$. (A) depicts the three basic open sets $\mathbb{D}_\mathcal{A}$, $\mathbb{D}_\mathcal{B}$, $\mathbb{D}_\mathcal{C}$ on $(\mathbb{R}\subset\mathbb{R}^2$): discs disjoint from $\mathbb{R}$ are governed by the braided tensor category structure on $\mathcal{A}$ and $\mathcal{B}$ in each connected region, while disks intersecting $\mathbb{R}$ are governed by the tensor structure of $\mathcal{C}$. (B) depicts an disks embedding $\mathbb{D}_\mathcal{A}\sqcup\mathbb{D}_\mathcal{B}\hookrightarrow \mathcal{D}_\mathcal{C}$; this induces a functor $F:\mathcal{A}\boxtimes\mathcal{B}\to\mathcal{C}$. (C) depicts an isotopy between two disk embeddings $\mathbb{D}_\mathcal{A}\sqcup\mathbb{D}_\mathcal{A}\sqcup\mathbb{D}_\mathcal{B}\sqcup\mathbb{D}_\mathcal{B}\to\mathbb{D}_\mathcal{C}$; this induces an isomorphism $J:F(-\otimes -) \xrightarrow{\sim} F(-)\otimes F(-)$, upgrading $F$ to a tensor functor. (D) depicts an isotopy between two disk embeddings $\mathbb{D}_\mathcal{A}\sqcup\mathbb{D}_\mathcal{C}\to\mathbb{D}_\mathcal{C}$; this induces a half-braiding on the image of $F$, and hence a lift $F:\mathcal{A}\to Z(\mathcal{C})$. The analogous half-braiding on $\mathcal{B}$ induces a lift $F:\mathcal{B}^{{bop}}\to Z(\mathcal{C})$, owing to the differing orientation of the bottom region relative to $\mathbb{R}$
• Figure 3: (A) depicts the basic open sets on the stratified space $(\mathbb{R}\sqcup \mathbb{R} \subset\mathbb{R}^2$): a locally constant factorization algebra $\mathcal{F}$ is defined by labelling the basic opens as indicated. (B) depicts a map, $\pi: (\mathbb{R}\sqcup\mathbb{R}\subset\mathbb{R}^2) \to (\mathbb{R}\subset\mathbb{R}^2)$, of stratified spaces collapsing the region between the two lines. The composition is defined as $\mathcal{C}\circ\mathcal{D} := \pi_*(\mathcal{F})$. Excision yields an equivalence $\mathcal{C}\circ\mathcal{D}\simeq \mathcal{C}\boxtimes_{\mathcal{A}_2}\mathcal{D}$, as categories, with structure maps given as in Proposition \ref{['prop:domain-wall-composition']}.
• Figure 4: (A) depicts the stratified space $(\mathbb{R}^0\subset \mathbb{R} \subset \mathbb{R}^2)$. (B) depicts the five basic open sets appearing in the stratification. (C) depicts an isotopy between two composite disk inclusions; this induces the central structure on the bimodule $\mathcal{M}$, identifying the action of $\mathcal{A}$ on $\mathcal{M}$ through $\mathcal{C}$ and through $\mathcal{D}$. The analogous isotopy on $\mathcal{B}$ gives the same structure for the induced $\mathcal{B}^{{bop}}$-action.

Theorems & Definitions (113)

• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10
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• Remark 1.13