Higher Morita Categories and Theories as Objects

Morita theory treats an algebra as an object, a bimodule as a morphism, and an intertwiner as a morphism between morphisms. This matches the dimensions of extended topological field theory: bulk phases, interfaces, and junction operators compose at different categorical levels. The construction requires relative tensor products to exist and behave coherently. A fully extended theory additionally requires duals and adjoints; an arbitrary algebra in the Morita category does not automatically supply them.

Required background. Locally Constant Factorization Algebras and Eₙ Algebras supplies the higher algebras attached to disks. Factorization Homology and Manifold Invariants supplies their integration over manifolds. Helpful background. Derived Intersections, Boundary Conditions, and Correspondences gives the gluing pattern for boundaries. QFT Frameworks, Object Classes, and Typed Maps prevents comparisons between unlike morphisms. Endomorphisms, Intertwiners, and Tensor Products supplies the operator-algebraic analogue.

The Morita bicategory and its composition law

Fix a field $k$. The elementary Morita bicategory $\operatorname{Alg}_2(k)$ has

• associative unital $k$-algebras as objects;
• $(A,B)$-bimodules as one-morphisms $A\to B$, with a fixed left–right convention;
• bimodule homomorphisms as two-morphisms.

If ${}_AM_B$ and ${}_BN_C$ are composable, their composite is

$${}_AM_B\otimes_B{}_BN_C.$$

The two triple composites are connected by the canonical associator

$$(M\otimes_BN)\otimes_CP \xrightarrow{\ \simeq\ } M\otimes_B(N\otimes_CP).$$

They are naturally isomorphic, not literally equal in a given presentation. The pentagon identity for this associator is the coherence condition that makes repeated interface fusion unambiguous.

In a monoidal infinity-category $\mathcal C$, relative tensor products are geometric realizations of bar constructions. Haugseng assumes “good relative tensor products,” which includes existence and preservation conditions needed for composition, and then constructs the double infinity-category whose underlying $(\infty,2)$-category has algebras, bimodules, and bimodule maps Haugseng 2017, Theorem 4.39, article p. 49. Iterating the construction organizes $E_n$ algebras and iterated bimodules into higher Morita categories. The result is conditional on those colimits and compatibility properties; it is not a formal consequence of writing $\otimes_B$.

Morita equivalence between $A$ and $B$ means an invertible one-morphism in this bicategory. It implies an equivalence of suitable module categories. It does not imply $A\cong B$ as algebras, and it need not preserve a chosen trace, topology, involution, or state unless those data are included in the objects and morphisms.

First application: finite-group topological gauge theory

The interfaces and point junctions described in Fusion, Junctions, and Endpoints have a finite algebraic model. Let $G$ be a finite group and let $k$ be a characteristic-zero field. Set

$$A=k[G],\qquad \lambda\!\left(\sum_{g\in G}a_g g\right)=a_e.$$

The bilinear form $(a,b)\mapsto\lambda(ab)$ is symmetric and nondegenerate. The separability idempotent is

$$e_A=\frac1{|G|}\sum_{g\in G}g\otimes g^{-1},$$

which satisfies $\mu(e_A)=1$ and is central for the left and right $A$ actions. Hence $A$ is a separable symmetric Frobenius algebra. In the algebra-valued description of oriented fully extended two-dimensional topological field theory, this is precisely the finiteness structure needed at a point: Schommer-Pries identifies such theories with separable symmetric Frobenius algebras and structure-preserving Morita equivalences 2011, Theorem 3.52, p. 230.

In this model, boundary conditions are finite $A$-modules, an interface from an $A$-phase to a $B$-phase is an $(A,B)$-bimodule, and a point defect between interfaces $M$ and $M'$ is an intertwiner

$$\operatorname{Hom}_{A\text{-}B}(M,M').$$

Fusion of an $(A,B)$ interface $M$ with a $(B,C)$ interface $N$ is $M\otimes_BN$. A junction of three interfaces is insensitive to parenthesization only after inserting the canonical associator. This gives the promised concrete hierarchy: algebras are phases, bimodules are codimension-one defects, and intertwiners are codimension-two junction operators.

An independent check is the trivial group. If $G=\{e\}$, then $A=k$, boundary conditions are vector spaces, interface fusion is ordinary tensor product over $k$, and the Frobenius trace is the identity. The construction collapses to the expected trivial two-dimensional theory.

The finite-group example is topological and finite. It does not establish that a general continuum gauge theory has a fully dualizable object in a specified analytic higher category. Nor does it say that every module is an admissible physical boundary condition; orientation, anomaly, unitarity, and locality constraints can select a smaller class.

Dualizability and the adversarial infinite algebra

Full extension down to points invokes duals at every morphism level. In $\operatorname{Alg}_2(k)$ over a field, a fully dualizable algebra is finite-dimensional and separable; an oriented structure further supplies a suitable nondegenerate symmetric trace. Schommer-Pries states the algebra, bimodule, and intertwiner levels explicitly and relates full dualizability to separability and finite projectivity 2011, Definition 3.70 and §3.8.4, p. 238.

Now replace $k[G]$ by $A=k[x_1,x_2,\ldots]$. Treating $A$ with finite semisimple formulas is the adversarial failure. As an $A\otimes A^{\mathrm{op}}$-module it is not finite projective in the required sense, the evaluation bimodule need not possess the adjoints demanded by full dualizability, and relative tensor products can require derived completions. The string-diagram manipulations of a fully extended theory are therefore not licensed.

This supplies two nonconverses. Having an associative fusion rule does not prove the existence of all adjoints and duals. Having equivalent module categories does not prove equality of traces, partition functions, or local junction observables. Each preserved datum must be named in the Morita equivalence.

Exercises

Verify the two defining properties of $e_A$ for $A=k[G]$.

Solution

Multiplication gives $\mu(e_A)=|G|^{-1}\sum_g gg^{-1}=1$. For $h\in G$,

$$(h\otimes1)e_A=|G|^{-1}\sum_g hg\otimes g^{-1} =|G|^{-1}\sum_{g'}g'\otimes g'^{-1}h=(1\otimes h)e_A.$$

after the substitution $g'=hg$. Linearity extends the identity to all $a\in A$.

Show that Morita equivalence need not be algebra isomorphism.

Solution

For $n>1$, the matrix algebra $M_n(k)$ is Morita equivalent to $k$: the column module $k^n$ and its dual implement inverse bimodules, and $M_n(k)$-modules are equivalent to vector spaces. But $M_n(k)$ and $k$ are not isomorphic as algebras because their dimensions differ. Thus the preserved object is the module theory, not the multiplication table itself.

References

• Haugseng 2017, The Higher Morita Category of Eₙ-Algebras, Geometry & Topology 21(3), 1631–1730. Open PDF
• Schommer-Pries 2011, The Classification of Two-Dimensional Extended Topological Field Theories, PhD thesis, University of California, Berkeley. Open PDF