Fully Extended TQFTs and Higher Categories

A fully extended $n$-dimensional TQFT assigns data not only to closed $(n-1)$-manifolds and $n$-bordisms, but to points, intervals, corners, and all strata through dimension $n$. These values live in a symmetric monoidal $(\infty,n)$-category: objects are point values, $1$-morphisms are interval values, and successive bordisms give successive morphisms, while morphisms above level $n$ are invertible coherences. Full extension demands duals and adjoints at every visible level. It is strictly stronger than an Atiyah–Segal theory on closed boundaries.

Required background. Bordism categories and symmetric monoidal TQFTs provide the unextended truncation; tensor products provide monoidality; and categories, functors, and natural transformations provide the first categorical level.

Helpful background. Fusion, junctions, and endpoints motivate why lower-dimensional strata carry physical data.

Bordisms down to points

Write $\operatorname{Bord}^{\xi}_n$ for the symmetric monoidal $(\infty,n)$-category of bordisms with tangential structure $\xi$. In a fully extended convention, its objects are structured $0$-manifolds, its $1$-morphisms are $1$-bordisms, and its $k$-morphisms for $k\le n$ are $k$-dimensional bordisms with corners and compatible collars. Diffeomorphisms, isotopies, and higher isotopies supply invertible higher morphisms. Disjoint union is monoidal.

A fully extended theory is a symmetric monoidal functor

$$Z:\operatorname{Bord}^{\xi}_n\longrightarrow\mathcal C$$

to a declared symmetric monoidal $(\infty,n)$-category $\mathcal C$. Its ordinary Atiyah–Segal truncation remembers only closed $(n-1)$-manifolds and $n$-bordisms. That truncation generally loses the point object and may identify distinct extended theories.

The hierarchy matters already in two dimensions:

$$\begin{array}{c|c|c} \text{bordism datum}&\text{categorical level}&\text{typical algebraic value}\\ \hline \text{oriented point}&\text{object}&\text{algebra }A\\ \text{interval}&1\text{-morphism}&\text{bimodule}\\ \text{surface with corners}&2\text{-morphism}&\text{bimodule map}. \end{array}$$

The oppositely oriented point is sent to a dual object. Cups and caps demand evaluation and coevaluation $1$-morphisms; two-dimensional handle moves require adjoints and coherent $2$-isomorphisms. Lurie formulates full dualizability by repeatedly discarding morphisms without left and right adjoints in Lurie 2009, Definition 2.3.21 and Warning 2.3.22, printed pp. 42–43.

The Morita two-category example

Let $\operatorname{Alg}_2(\mathbb C)$ have finite-dimensional complex algebras as objects, finite-dimensional bimodules as $1$-morphisms, and bimodule homomorphisms as $2$-morphisms. Tensor product supplies the symmetric monoidal structure. A two-dimensional extended theory may assign

$$Z(+)=A, \qquad Z(-)=A^{\mathrm{op}}.$$

An interval between labels is assigned an appropriate module or bimodule. Gluing intervals is relative tensor product:

$${}_A M_B\circ{}_B N_C \longmapsto M\otimes_BN.$$

A corner at which interval bordisms meet becomes an intertwiner between such tensor products. The circle value is not the point algebra $A$ itself. For the regular defect it is a trace-like object, modeled by Hochschild homology

$$Z(S^1)\simeq HH_0(A)=A/[A,A]$$

in the underived finite-dimensional setting. For $A=M_m(\mathbb C)$ this is one-dimensional even though $A$ has dimension $m^2$. That computation independently shows why unextended circle data cannot simply be relabeled as a point value.

The exact first application refines the two-dimensional oriented theory used in state spaces, cobordisms, and gluing: place an algebra on a point, bimodules on intervals, and intertwiners on surfaces with corners. In the oriented Morita target, the full classification also requires separability and symmetric Frobenius trace data; Schommer-Pries proves this generators-and-relations classification in Schommer-Pries 2014, Chapter 3, especially §3.5.

Why adjoints appear

The proof mechanism is geometric. Turning a bordism around reverses its categorical direction. A birth or death of a pair of strata produces units and counits for an adjunction. Isotopies that cancel a cup against a cap become triangle identities. At the next level, different cancellation movies must themselves be coherently equivalent. Consequently, extending one level deeper is possible only when the corresponding morphisms have left and right adjoints.

An independent check uses the negative point. The composites

$$A\longrightarrow A\otimes A^{\mathrm{op}}\otimes A \longrightarrow A$$

must be equivalent to the identity through the regular evaluation and coevaluation bimodules. At the interval level, the same check is the unit law for relative tensor products.

The adversarial failure chooses a point object with no dual, or a bimodule with no adjoint, and nevertheless asks for a fully extended functor. The missing geometric bordism is concrete: without a dual, the cap or cup at the point has no image; without an adjoint, the corresponding fold of an interval cannot be assigned. An unextended closed-surface functor may still exist, but the claimed extension to points and corners does not.

Exercises

Compute $HH_0(M_m(\mathbb C))$.

Solution

Every traceless matrix is a sum of commutators, while the ordinary trace vanishes on commutators. Hence the quotient by $[A,A]$ is represented by scalar matrices and is isomorphic to $\mathbb C$.

Why are higher morphisms above dimension $n$ invertible?

Solution

The geometric theory has no noninvertible bordisms of dimension above $n$. What remains are equivalences between bordisms—diffeomorphisms, isotopies, and their coherences—so the target is an $(\infty,n)$-category rather than an unrestricted $\infty$-category with noninvertible morphisms at every level.

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.