Categorical Classification: Equivalence and Completeness Limits

A categorical classification is meaningful only after naming both the input data and the equivalence relation. Equality of fusion rings, monoidal equivalence, Morita equivalence, braided equivalence of centers, equivalence of fully extended TQFTs, and equivalence of local QFTs are progressively different claims. Categorical data can classify a bounded topological problem while omitting symmetry enrichment, tangential structure, positivity, local observables, and analytic dynamics.

Required background. Fusion Categories, Module Categories, and Bimodule Defects supplies Morita equivalence. Helpful background. Gauging, Equivariantization, Orbifolds, and Condensation supplies completion operations, Generalized-Symmetry Sectors, Selection Rules, and Reconstruction supplies realization limits, Extended-TQFT Classification: Scope and Counterexamples supplies the cobordism-hypothesis boundary, and Conformal-Net Classification: Invariants and Limits supplies an analytic QFT contrast.

A hierarchy of equivalences

Let $\mathcal C$ and $\mathcal D$ be fusion categories. The following questions retain different information.

1. An isomorphism $K_0(\mathcal C)\cong K_0(\mathcal D)$ matches simple labels and fusion coefficients. It forgets associators and pivotal, braided, and unitary structures.
2. A monoidal equivalence matches objects, morphisms, tensor product, unit, and associator coherently. It need not preserve a specified braiding or dagger.
3. A Morita equivalence is an invertible $(\mathcal C,\mathcal D)$-bimodule category. It identifies their bulk centers while allowing different boundary presentations.
4. A braided equivalence $Z(\mathcal C)\simeq Z(\mathcal D)$ identifies bulk anyon categories under the fusion-category hypotheses. Etingof, Nikshych, and Ostrik prove that the center functor is fully faithful on the Brauer–Picard $2$-groupoid in Etingof, Nikshych, and Ostrik 2010, Theorem 1.1 and §5, printed pp. 3–5 and 27–32 (PDF).
5. An equivalence of fully extended TQFTs must also preserve the chosen bordism and tangential structures and the full dualizable target data.
6. Equivalence of physical QFTs requires the relevant local observable algebras or correlators, state spaces, positivity, and dynamics—not merely their topological defect subsector.

Every upward inference needs a theorem. A center may be a complete invariant of Morita equivalence in the stated fusion setting while still being incomplete for enriched phases or non-topological QFTs.

Morita data with different enrichment

Let $\mathcal C=\mathrm{Vec}_{\mathbb Z_3}$, the pointed fusion category of $\mathbb Z_3$-graded vector spaces. Equip it with an external $K=\mathbb Z_2$ symmetry in two ways. In the first, $K$ acts trivially. In the second, its generator acts by inversion $a\mapsto-a$ on $\mathbb Z_3$ grades. The underlying fusion category—and hence its Morita class and Drinfeld center—is the same in both cases.

The enriched data are nevertheless inequivalent. In the inversion action, the two nontrivial simple objects are exchanged; in the trivial action, each is fixed. Equivariantization therefore has different orbit and stabilizer data, and symmetry defects act differently on boundaries. Forgetting $K$ identifies the phases at the un-enriched categorical level; requiring a $K$-equivariant equivalence keeps them distinct.

This example realizes the exact first application returned to Non-Abelian Topological Orders: two phases may share Morita-equivalent fusion data while differing in symmetry enrichment. The mathematical comparison states the level at which they agree. The many-body treatment decides whether the enriched phases can be connected without breaking symmetry or closing a gap.

An independent diagnostic examines orbit sizes. The trivial action has three fixed simples. Inversion has one fixed simple and one orbit of size two. No relabeling intertwines these $K$-actions, even though the un-enriched fusion table is identical.

Completeness and realization

A proposed classification map

$$\mathfrak F:\{\text{field theories}\}/\simeq_{\mathrm{phys}} \longrightarrow \{\text{categorical invariants}\}/\simeq_{\mathrm{cat}}$$

must be tested for injectivity, surjectivity, and a well-defined domain. A nontrivial kernel means distinct theories share the invariant. Failure of surjectivity means some abstract category has no realization satisfying locality or positivity. A domain mismatch occurs when the theorem classifies fully extended topological functors but the claim concerns a gapless or non-topological QFT.

The cobordism hypothesis classifies fully extended TFTs valued in a chosen symmetric monoidal higher category by fully dualizable objects with the relevant homotopy-fixed-point data. It does not assert that every local QFT is topological or fully extended. Likewise, a defect category extracted from a QFT can forget stress tensors, scaling dimensions, continuous couplings, and non-topological operators.

Failure tests for categorical overclaims

Matching Grothendieck rings does not match associators: distinct $3$-cocycles can twist $\mathrm{Vec}_G$ while preserving the same fusion coefficients. Matching centers establishes the appropriate Morita statement but does not preserve a chosen boundary, pivotal normalization, or symmetry action. Matching a fully dualizable object classifies the functor only in the specified target and tangential setting.

The adversarial claim declares two QFTs equivalent because their fusion rings or Drinfeld centers agree. The $\mathbb Z_3$ example defeats the enrichment claim, while continuous changes of non-topological couplings defeat reconstruction of local dynamics. The strongest surviving conclusion must name the exact quotient: equality of fusion rings, Morita equivalence of fusion categories, or equivalence of a specified TQFT. It cannot be silently upgraded to physical QFT equivalence.

Exercises

Why does an isomorphism of Grothendieck rings fail to determine the associator?

Solution

The Grothendieck ring records only multiplicities in tensor-product decompositions. Associators are natural isomorphisms between different parenthesizations and can be twisted by nontrivial cohomology classes while leaving all multiplicities unchanged. Hence identical fusion coefficients can support inequivalent monoidal categories.

References

• Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. “Fusion Categories and Homotopy Theory.” Quantum Topology 1 (2010): 209–273. DOI; Open PDF.
• Lurie, Jacob. “On the Classification of Topological Field Theories.” In Current Developments in Mathematics 2008, 129–280. Somerville, MA: International Press, 2009. Open PDF.