Dualizability and the Cobordism Hypothesis

The framed cobordism hypothesis classifies fully extended $n$-dimensional framed TQFTs valued in a specified symmetric monoidal $(\infty,n)$-category by fully dualizable objects of that target. Evaluation at a positively framed point is an equivalence of $\infty$-groupoids, not merely a map on isomorphism classes. The theorem does not classify unextended theories, oriented theories without fixed-point data, nondualizable targets, or arbitrary physical QFTs.

Required background. Fully extended TQFTs fix the domain and extension depth, while higher-categorical targets and dualizability levels fix the target-side finiteness and adjoint conditions.

Helpful background. Bordism and tangential structures explain why a framing is extra geometric data rather than a synonym for orientation.

The framed classification theorem

Let $\mathcal C$ be a symmetric monoidal $(\infty,n)$-category, and let $\mathcal C^{\mathrm{fd}}$ denote its fully dualizable part. Write $(\mathcal C^{\mathrm{fd}})^\sim$ for the maximal $\infty$-groupoid obtained by retaining objects and equivalences. The framed cobordism hypothesis states that evaluation at the positively framed point induces an equivalence

$$\operatorname{ev}_{*}: \operatorname{Fun}^{\otimes} \left(\operatorname{Bord}^{\mathrm{fr}}_n,\mathcal C\right) \xrightarrow{\;\simeq\;} (\mathcal C^{\mathrm{fd}})^\sim.$$

Equivalently, the fully extended framed bordism category is freely generated, as a symmetric monoidal $(\infty,n)$-category with duals, by one object. The precise statement is Lurie 2009, Theorem 2.4.6 and Remarks 2.4.7–2.4.9, printed pp. 43–44.

Every noun limits the result. The dimension $n$ fixes how many adjoint levels full dualizability contains. “Framed” fixes the tangential structure. “Fully extended” means values down to points. The target $\mathcal C$ fixes the algebraic meaning of objects and equivalences. The right-hand side is not all of $\mathcal C$, and changing $\mathcal C$ changes the classification.

Why the value on a point is enough

The theorem’s mechanism is a higher-dimensional generators-and-relations argument. A Morse function decomposes a bordism into handles. Index-zero and index-$n$ handles are coevaluation and evaluation for object duality. Intermediate handles give units and counits for adjunctions of lower morphisms. Handle cancellation is the triangle identity; changes of Morse function become higher coherence relations. Full dualizability supplies exactly the algebraic images of these generators and relations.

This sketches the architecture, not a replacement for the proof: controlling families of Morse functions, corners, and all coherences requires the $(\infty,n)$-categorical construction. The nontrivial part is that the resulting assignments are unique up to a contractible space of choices once the fully dualizable point object is fixed.

An important consequence is that equivalence is the correct comparison. If two point objects are equivalent in $(\mathcal C^{\mathrm{fd}})^\sim$, the associated framed theories are equivalent as symmetric monoidal functors. The theorem does not say that different presentations of those objects are literally equal.

Two-dimensional Morita application

Take $\mathcal C=\operatorname{Alg}_2(\mathbb C)$, the Morita $2$-category of finite-dimensional algebras, bimodules, and intertwiners. Its fully dualizable objects are finite-dimensional separable algebras, hence semisimple algebras over $\mathbb C$. Therefore a semisimple algebra $A$ determines a framed fully extended two-dimensional TQFT.

For example,

$$A=M_r(\mathbb C)$$

is fully dualizable. Its dual is $A^{\mathrm{op}}$, evaluation is the regular bimodule, and separability gives both adjoints. Since $M_r(\mathbb C)$ is Morita equivalent to $\mathbb C$, their point objects are equivalent in the Morita target and the corresponding framed theories are equivalent. This does not mean the algebras are isomorphic when $r>1$.

This is the exact first application passed to state spaces, cobordisms, and gluing: classify framed fully extended two-dimensional theories in the Morita target by fully dualizable finite-dimensional separable algebras. Schommer-Pries independently presents the two-dimensional bordism bicategory by generators and relations and obtains the corresponding algebraic classification in Schommer-Pries 2014, Chapter 3.

The missing-adjoint counterexample

The dual-number algebra

$$D=\mathbb C[\epsilon]/(\epsilon^2)$$

has an object dual $D^{\mathrm{op}}$ in the Morita $2$-category. It is even a commutative Frobenius algebra, so it defines an unextended oriented two-dimensional TQFT. But it is not separable: its nilpotent radical is nonzero. The regular evaluation bimodule lacks the adjoints required for full $2$-dualizability. Therefore $D$ does not lie in $(\mathcal C^{\mathrm{fd}})^\sim$ and cannot define the claimed fully extended Morita-valued theory.

An independent check distinguishes the circle and point. For $M_r(\mathbb C)$, the trace construction gives $HH_0(A)\cong\mathbb C$ on the circle, consistent with Morita equivalence to $\mathbb C$. The point still carries the matrix algebra object. This confirms that the theorem reconstructs all strata coherently rather than identifying their values.

The adversarial failure is to verify only an object dual and quote the theorem. The strongest surviving statement is $1$-dualizability. Without adjoints for evaluation at every lower level, the point does not classify a fully extended theory.

Exercises

Why does Morita equivalence, rather than algebra isomorphism, appear?

Solution

Equivalences between objects of the Morita $2$-category are invertible bimodules. Such a bimodule identifies module categories and has an inverse under relative tensor product. This is precisely Morita equivalence, which is weaker than algebra isomorphism.

What additional datum is needed to pass from framed to oriented?

Solution

One needs a coherent $SO(n)$ homotopy fixed point for the natural $O(n)$ action on the fully dualizable object. In two dimensions this includes a trivialization of the Serre automorphism and is represented by suitable Calabi–Yau or symmetric Frobenius trace data in the Morita example.

References

• Lurie, Jacob. “On the Classification of Topological Field Theories.” In Current Developments in Mathematics 2008, 129–280. Somerville, MA: International Press, 2009. Open PDF.
• Schommer-Pries, Christopher J. The Classification of Two-Dimensional Extended Topological Field Theories. PhD thesis, University of California, Berkeley, 2009; expanded version 2014. Open PDF.