Higher-Categorical Targets and Levels of Dualizability

The extension depth of a TQFT is limited by the adjoints available in its chosen target. Object duality is only the first level: a two-dimensional fully extended theory also requires left and right adjoints for evaluation and coevaluation $1$-morphisms, and higher dimensions continue this pattern. In the Morita $2$-category of algebras, bimodules, and intertwiners, finite-dimensional separable algebras are fully dualizable; a finite-dimensional Frobenius algebra need not be. The target and categorical height are therefore part of every classification theorem.

Required background. Fully extended TQFTs define the stratified assignment, while the map of derived, higher, and factorization frameworks distinguishes the categorical models being used.

Helpful background. Higher Morita categories supply the general target construction, and operators, boundaries, and relative theories motivate its morphisms.

Levels of dualizability

Let $\mathcal C$ be a symmetric monoidal $(\infty,n)$-category. An object $X$ is $1$-dualizable if there is an object $X^\vee$ and morphisms

$$\operatorname{coev}:\mathbb 1\to X\otimes X^\vee, \qquad \operatorname{ev}:X^\vee\otimes X\to\mathbb 1$$

satisfying the two snake identities up to coherent equivalence. It is $2$-dualizable if, in addition, the evaluation and coevaluation $1$-morphisms have both left and right adjoints. Iterating the requirement through $n$ gives full dualizability. More invariantly, the fully dualizable subcategory is obtained by retaining dualizable objects, then only morphisms with two-sided adjoints at every successive level.

For $n=2$, it is enough to check that $X$ has a dual and that $\operatorname{ev}_X$ has both left and right adjoints; the remaining adjoints follow. This criterion and its proof via the Serre automorphism appear in Lurie 2009, Proposition 4.2.3, printed pp. 91–92. The qualifier “in an $(\infty,2)$-category” is essential: the same object regarded in a taller target faces additional adjoint conditions.

Algebras, bimodules, and separability

Consider $\operatorname{Alg}_2(k)$ for a field $k$. Its objects are finite-dimensional associative $k$-algebras, a $1$-morphism $A\to B$ is a suitably oriented $(B,A)$-bimodule, and $2$-morphisms are bimodule maps. Composition is relative tensor product. The monoidal dual of $A$ is $A^{\mathrm{op}}$, with the regular bimodule providing evaluation:

$${}_{k}A_{A\otimes A^{\mathrm{op}}}: A\otimes A^{\mathrm{op}}\longrightarrow k.$$

Thus object duality is widespread. Full $2$-dualizability asks whether this regular bimodule has adjoints as a bimodule. That occurs, under the finite projectivity assumptions in this target, precisely when multiplication

$$\mu:A\otimes_k A^{\mathrm{op}}\longrightarrow A$$

splits as an $A$-bimodule map. This is separability. Over $\mathbb C$, a finite-dimensional algebra is separable exactly when it is semisimple.

For example, $A=M_m(\mathbb C)$ is separable. A separability idempotent is

$$p=\frac1m\sum_{i,j=1}^{m}E_{ij}\otimes E_{ji},$$

which satisfies $\mu(p)=1$ and $(a\otimes1)p=p(1\otimes a)$. It constructs the adjoint data for the regular bimodule. Finite direct sums of matrix algebras behave similarly.

By contrast, the dual-number algebra

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

is finite-dimensional and Frobenius: the functional $\lambda(1)=0$, $\lambda(\epsilon)=1$ makes $\lambda(ab)$ nondegenerate. But $D$ is not semisimple and hence not separable. It can define an unextended oriented two-dimensional TQFT because it is commutative Frobenius, yet it cannot be promoted to a fully extended theory valued in this Morita $2$-category. This is the cleanest nonconverse.

The exact first application is returned to fusion, junctions, and endpoints: place algebras, bimodules, and intertwiners in the Morita target, then select the separable Frobenius algebras whose regular evaluation morphisms are adjointable. The low-dimensional definition of $k$-full dualizability is given in Schommer-Pries 2013, Definition 7.5, printed p. 22.

Adjoints as the extension checkpoint

The proof mechanism translates handle cancellation into triangle identities. A cup/cap for an object gives its dual. A fold of a $1$-bordism gives the unit or counit of an adjunction for the corresponding bimodule. A geometric cancellation of two folds forces the triangle identity. If the bimodule is not finitely generated projective on the required side, its formal linear dual need not define an adjoint under relative tensor product.

An independent check for $M_m(\mathbb C)$ multiplies the separability idempotent and checks centrality as above. For $D$, its nonzero nilpotent ideal $(\epsilon)$ lies in the Jacobson radical, so semisimplicity—and therefore separability over $\mathbb C$—fails.

The adversarial failure takes ordinary object duals as sufficient for every higher adjoint. The algebra $D$ has the object dual $D^{\mathrm{op}}$ but its evaluation bimodule lacks the required two-sided adjoints. The strongest surviving conclusion is $1$-dualizability in the chosen target, not a fully extended two-dimensional TQFT.

Exercises

Verify the centrality property of $p$ for a matrix unit $E_{ab}$.

Solution

Using $E_{ab}E_{ij}=\delta_{bi}E_{aj}$ and $E_{ji}E_{ab}=\delta_{ia}E_{jb}$, both $(E_{ab}\otimes1)p$ and $p(1\otimes E_{ab})$ reduce, after relabeling, to $m^{-1}\sum_jE_{aj}\otimes E_{jb}$.

Why does a Frobenius pairing not imply separability?

Solution

A Frobenius pairing identifies $A$ with its linear dual as a module. Separability asks for multiplication to split as an $A$-bimodule map, a stronger condition that excludes nilpotent radical directions. The dual numbers supply the explicit counterexample.

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. “Dualizability in Low-Dimensional Higher Category Theory.” In Topology and Field Theories, Contemporary Mathematics 613, 111–176. Providence, RI: American Mathematical Society, 2014. DOI; Open PDF.