Fusion 2-Categories: Algebra & TQFT

Updated 3 August 2025

Fusion 2-categories are finite semisimple, additive, rigid monoidal 2-categories with duals that generalize fusion categories to higher dimensions.
They feature a homotopy-coherent classification that connects braided fusion 1-categories with group cohomological data, underpinning extended TQFTs and quantum symmetries.
Their robust Morita theory and dualizability support state-sum invariants and modeling of non-invertible symmetries in topological phases of matter.

A fusion 2-category is a higher-categorical generalization of a fusion category, providing a semisimple, additive, rigid monoidal 2-category structure in which every object has duals and the monoidal unit is simple. Fusion 2-categories serve as the natural algebraic framework for describing the fusion and interactions of extended (e.g., codimension-2) operators in mathematical physics, especially in the context of extended topological quantum field theory (TQFT), higher representation theory, and topological phases of matter in dimensions three and four. Their theory amalgamates ideas from semisimple 2-category theory, module and bimodule categories, group cohomology, and the higher-categorical analogs of Morita theory and the Brauer–Picard group. Recent advances have established a precise and homotopy-coherent classification, clarified their pivotal role in field theory, and made explicit connections to state-sum invariants, quantum symmetries, and categorical dualities.

1. Formal Structure and Definition

A fusion 2-category is a finite semisimple, additive, $\mathbb{C}$ -linear, monoidal 2-category $\mathcal{C}$ in which:

Every object possesses both left and right duals.
The monoidal unit $I$ is simple (i.e., indecomposable and not a direct sum).
Each Hom-category $\operatorname{Hom}_\mathcal{C}(A,B)$ is a finite semisimple (linear) category.
The monoidal product functor $\mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is (bi)additive and (possibly weakly) associative.
Morphisms and 2-morphisms are (complex)-linear and direct sums (as well as splitting of idempotents) exist at every level.

Dualizability of both objects and 1-morphisms plays a crucial role: for each $A \in \mathcal{C}$ , there exists $A^\# \in \mathcal{C}$ and adjunction data $(e_A, i_A)$ so that $(A, A^\#)$ are dual in the 2-categorical sense. The idempotent completeness and additive/semisimple structure ensure that the 2-category is equivalent to a 2-category of module categories over a multifusion 1-category (see (Douglas et al., 2018, Décoppet, 2021, Décoppet, 2022, Décoppet, 2022)).

The core algebraic object governing a fusion 2-category is the braided fusion 1-category $\Omega\mathcal{C} := \operatorname{End}_\mathcal{C}(I)$ , also called the “loop” or endomorphism category of the unit. Its symmetric center $\mathscr{Z}_2(\Omega\mathcal{C})$ is a symmetric fusion category, typically Tannakian (e.g., $\operatorname{Rep}(H)$ for finite $H$ ) in the bosonic case or super-Tannakian (e.g., $\mathbf{SVect}$ , $\operatorname{Rep}(H, z)$ ) in the fermionic case (Décoppet et al., 8 Nov 2024).

2. Classification and Homotopy Coherence

The complete classification of fusion 2-categories is achieved via a homotopy-coherent dictionary incorporating braided fusion 1-categories and group cohomological data (Décoppet et al., 8 Nov 2024). Given a fusion 2-category $\mathcal{F}$ , its equivalence class up to monoidal equivalence is uniquely determined by:

A nondegenerate braided fusion 1-category $A$ (typically the de-equivariantization of $\Omega\mathcal{F}$ ).
A finite group or supergroup inclusion $H \subset G$ corresponding to the symmetric centers ( $\mathscr{Z}_2(\Omega\mathcal{F}) \simeq \operatorname{Rep}(H)$ , $\Omega\mathscr{Z}(\mathcal{F}) \simeq \operatorname{Rep}(G)$ ).
A monoidal functor $\rho: H \to \operatorname{Aut}^{\otimes, \mathrm{br}}(A)$ representing a braided autoaction.
An obstruction or “anomaly” class $\pi \in H^4(BG, \mathbb{C}^\times)$ (bosonic), or a supercohomology class $\varpi \in \mathrm{SH}^{4+\omega}(BG)$ (fermionic, incorporating a graded action), together with a prescribed homotopy between the 3-cocycle representing the associativity anomaly of $\rho$ and $\pi|_H$ .

The homotopy-coherent structure is formalized as a 3-groupoid equivalence between multifusion 2-categories and certain commuting squares (Delphic squares) in the category of $\mathrm{B}\mathbb{Z}/2$ -equivariant spaces, encoding the symmetry and super-symmetry (see (Décoppet et al., 8 Nov 2024), Theorems A/B). This classification matches fusion 2-categories to “ $(A,H,G,\pi)$ ” data modulo homotopy, with analogous statements for the fermionic (“emergent fermion”) case.

3. Morita Theory, Dualizability, and Centers

Morita theory provides a robust categorical framework for fusion 2-categories, generalizing the role of module categories in fusion 1-categories (Décoppet, 2022). The essential elements include:

Bimodule 2-categories: For a fusion 2-category $\mathcal{C}$ , the 2-category of bimodules over a separable algebra object in $\mathcal{C}$ models Morita theory. The relative 2-Deligne tensor product $\mathcal{M} \boxtimes_\mathcal{C} \mathcal{N}$ composes left and right module 2-categories.
Morita equivalence: Two fusion 2-categories are Morita equivalent if their categories of module 2-categories are equivalent (as 3-categories) or, equivalently, if there exists an invertible bimodule 2-category implementing an equivalence between them (Décoppet, 2022).
Dualizability: Every fusion 2-category over an algebraically closed field of characteristic zero is fully dualizable in the symmetric monoidal Morita 4-category whose 1-morphisms are separable bimodule 2-categories and their higher morphisms (Décoppet, 2023). This supports the role of fusion 2-categories as target objects for fully extended 4d TQFTs via the cobordism hypothesis (see also (Décoppet, 2022)).
Drinfeld centers: The Drinfeld center $\mathscr{Z}(\mathcal{C})$ of a fusion 2-category is a finite semisimple braided monoidal 2-category invariant under Morita equivalence. When $\mathcal{C}$ is connected, Morita equivalence classes correspond precisely to Witt equivalence classes of underlying braided fusion 1-categories with shared symmetric centers (Décoppet, 2022).

4. Graded Extensions, Fermionic and Group-Theoretical Examples

Fusion 2-categories support a rich extension theory encompassing group gradings, supercohomology, and defect sectors:

Group-theoretical fusion 2-categories: For $G$ a finite group, subgroups $H$ , and 4-cocycles $\pi$ , the group-theoretical fusion 2-category $Bimod_{2Vect_G^\pi}(Vect_H^\psi)$ realizes “gauged” categories where bimodule sectors are labeled by double cosets $H \backslash G / H$ and carry fusion rules governed by the group data and cocycles (Décoppet et al., 2023).
Extension theory and fermionic fusion 2-categories: In the fermionic setting, the classification of strongly fusion 2-categories (with $\Omega\mathcal{C} = \mathbf{SVect}$ ) reduces to the homotopy-theoretic data $(G, w, \pi)$ , where $w \in H^2(BG; \mathbb{Z}/2)$ is a twist and $\pi \in \mathrm{SH}^{4+w}(BG)$ is a twisted supercohomology class (Décoppet, 5 Mar 2024). The Postnikov tower of the Brauer–Picard space for $\mathbf{SVect}$ governs these extensions and their topological defects.
Tambara–Yamagami fusion 2-categories: These are $\mathbb{Z}/2$ -graded fusion 2-categories whose neutral part is 2-vector spaces graded by a finite group and whose nontrivial part is Morita equivalent to $\mathbf{2Vect}$ . Their Morita theory, defect structure, and classification are governed by explicit group-theoretical and cohomological data (Décoppet et al., 2023).

5. State-Sum Invariants and Topological Field Theory

Fusion 2-categories feature prominently in constructing higher-dimensional state-sum invariants and TQFTs:

State-sum invariants for 4-manifolds: Given a spherical fusion 2-category $\mathcal{C}$ , the state-sum invariant $Z_\mathcal{C}(M)$ of a closed oriented PL 4-manifold $M$ is computed via a sum over labelings of the 1- and 2-simplices by simple objects and 1-morphisms, with contributions (quantum dimensions, 10j-symbols) encoding the higher categorical associativity and pivotal structure (Douglas et al., 2018).
Symmetry TFTs and fusion surface models: Fusion 2-categories can model non-invertible and higher-form symmetries in (2+1)d and (3+1)d lattice systems, generalizing anyon chains (1+1)d. Their use as input data yields fusion surface models whose topological order and symmetry content reflect the underlying 2-categorical structure, with defects and dualities classified by module and bimodule 2-categories (Inamura et al., 2023, Eck, 24 Jan 2025).

6. Finiteness, Rigidity, and Applications

The recent classification implies stringent finiteness and rigidity constraints:

Rank finiteness: There are only finitely many fusion 2-categories (up to equivalence) of a given rank (number of simple objects times the number of simple 1-morphisms in the endomorphism fusion 1-category), mirroring results in fusion 1-category theory.
Ocneanu rigidity: Fusion 2-categories admit no nontrivial first-order deformations; any family of such categories with fixed underlying data is necessarily constant (Décoppet et al., 8 Nov 2024).
Applications: Fusion 2-categories are used in classifying extended 4d TQFTs, modeling categorical and non-invertible symmetries, constructing state-sum invariants, and the paper of dualities and phases in topological phases of matter and condensed matter systems.

7. Outlook and Open Problems

Future research directions involve:

Extension of the classification and structure theory to positive characteristic and more general higher categories.
Concrete computations of extension and cohomological invariants in specific fusion 2-categories, especially in the fermionic and supercohomological settings (Décoppet, 5 Mar 2024).
Further development of the Morita and Witt equivalence frameworks and their consequences for higher TQFTs, categorical symmetries, and topological order.
Deepening the interplay between the algebraic (e.g., group-cohomological) classification and geometric/topological quantum field theory constructions, including exploration of higher braided structures and categorified centers.

The theory of fusion 2-categories thus provides a central organizing principle in modern higher representation theory, algebraic topology, and topological quantum field theory, connecting categorical algebra, group cohomology, and the emergent field of higher quantum symmetries.