Unitary 2-Category of Finite Type Hilbert Bimodules

• The topic is a higher categorical framework uniting C*-algebras, finite type Hilbert bimodules, and unitary intertwiners to rigorously analyze operator algebra phenomena.
• It enables a precise study of Morita equivalence, duality, and quantum symmetries through finite indices, finite orthonormal bases, and structured tensor products.
• The framework’s analytic and categorical models have practical applications in topological quantum field theory, state-sum constructions, and quantum-classical correspondence.

A unitary 2-category of finite type Hilbert bimodules is a higher categorical structure in which the objects are C*-algebras (or suitable generalizations), the 1-morphisms are Hilbert C*-bimodules of “finite type” (with properties such as finite index, compact generation, or admitting a finite orthonormal basis), and the 2-morphisms are unitary intertwiners (bounded adjointable bimodule maps). These 2-categories serve as a rigorous backdrop for the paper of Morita equivalence, quantum symmetries, operator algebraic dualities, and modular tensor constructions, and encapsulate both analytic and categorical features central to quantum algebra, operator algebras, and topological quantum field theory. The notion of “finite type” is motivated by analogies with fusion categories, projective 2-representations, and rigid module categories, ensuring dualizability, semisimplicity, and the transfer of structural properties.

1. Structural Definition and Foundational Properties

A unitary 2-category consists of the following data and properties:

• Objects: C*-algebras, possibly finite direct sums of matrix algebras (e.g., $M = \ell^\infty\left(\bigoplus_{i\in I} M_i\right)$ with each $M_i$ a finite-dimensional matrix algebra), groupoid quantales, or rigid C*-tensor categories (depending on the context) (Chen et al., 7 Oct 2024, Rollier, 6 May 2024, Quijano et al., 2018).
• 1-Morphisms: Finite type Hilbert C*-bimodules or bimodule categories with additional structure; these may be realized as:
  • Imprimitivity bimodules of finite index (Forough et al., 2016),
  • Hilbert bimodules with finite orthonormal basis (Aaserud et al., 2019),
  • Objects in module 2-categories for H*-algebras or Q-systems in a unitary multifusion category (Chen et al., 7 Oct 2024),
  • Bimodules associated to groupoid quantales with Hilbert basis (Quijano et al., 2018).
• 2-Morphisms: Unitary (adjointable) intertwiners between bimodules or functors, equipped with a dagger structure (involution) compatible with the module structure (Chen et al., 7 Oct 2024, Ferrer et al., 2019).
• Tensor Composition: Composition of 1-morphisms is given by the (interior) relative tensor product of Hilbert bimodules (or the Deligne product for bimodule categories) with associativity and unitarity constraints respected (Chen et al., 7 Oct 2024, Giorgetti et al., 2017).
• Duality and Adjunctions: Duals (adjoints) exist: every 1-morphism has a left and right adjoint, implemented by conjugate bimodules (“rigidity”) (Chen et al., 7 Oct 2024, Décoppet, 2022).
• Positivity and Trace: A spherical weight or categorical trace is present, ensuring each hom-category is a (semisimple) 2-Hilbert space and that the C*-structure is compatible with direct sum decompositions (Hilbert direct sums) (Chen et al., 7 Oct 2024).
• Finite Type Conditions: Finiteness is imposed as either a uniform bound on the size/dimension of hom-categories (e.g., dimensions of centers of linking algebras), existence of finite orthonormal bases, or index-theoretic properties (e.g., finite Jones or numerical index; see key definitions in (Forough et al., 2016, Chen et al., 7 Oct 2024, Aaserud et al., 2019, Décoppet, 2022)).

2. Categorical Completions, Module Theory, and Idempotents

The construction and paper of these 2-categories requires several categorical and analytic completions:

• Hilbert Direct Sum Completion: To ensure closure under direct sums and idempotent splitting, one forms the Hilbert direct sum completion, where a direct sum of objects a₁ ⊕ a₂ has the associated endomorphism 1₍ₐ₁⊕ₐ₂₎ and trace split as Ψ₍ₐ₁⊕ₐ₂₎(f) = Ψ₍ₐ₁₎(f₁) + Ψ₍ₐ₂₎(f₂) (Chen et al., 7 Oct 2024).
• H*-Algebras and Module Categories: The unitary 2-category is described equivalently by H*-algebras (i.e., C*-Frobenius algebras with standardness conditions) and their bimodules, or by unitary module categories with trace in a unitary multifusion category ("H*-multifusion category") (Chen et al., 7 Oct 2024, Aaserud et al., 2019).
  • These admit a Morita theory: every module category is equivalent (up to isometry) to the category of modules over an H*-algebra, and returning, every H*-algebra gives rise to a module category (see equivalence Mod†(𝒞) ≅ H*Alg(𝒞)).
  • Idempotents split orthogonally and every module decomposes as a direct sum of simple units.
• Splitting of H*-Monads: Every H*-monad splits in the idempotent-completed 3-Hilbert space; this is essential for full semisimplicity and dualizability in the higher categorical framework (Chen et al., 7 Oct 2024).

3. Analytic and Algebraic Models: Operator Algebraic and Tensor Categorical Realizations

Multiple concrete models instantiate the unitary 2-category of finite type Hilbert bimodules:

• Operator Algebraic Realizations: Via Tomita bimodules and Fock space techniques, any rigid C*-tensor category (with simple unit) is realized as finite index endomorphisms and bimodules of a von Neumann factor (e.g., a free group factor) (Giorgetti et al., 2017).
  • The explicit construction: objects map to finite-index endomorphisms (or correspondences by Connes' picture) and tensor composition matches the interior tensor product.
  • Positivity and *-structures are preserved, and the dualities/dagger involutions are implemented analytically.
• Hilbert Module Categories: The category TLJ(d) (the Temperley-Lieb-Jones C*-tensor category) is equivalent (as a braided C*-tensor category) to the category of right Hilbert B-modules with a finite orthonormal basis, as shown via a diagrammatic construction and functorial isometries (Aaserud et al., 2019).
  • The passage from fusion rules to K₀(B) categorifies the correspondence between fusion rings and K-theory.
  • Similar analytic techniques extend to any rigid finitely generated braided C*-tensor category.
• Quantale-Theoretic and Sheaf-Theoretic Models: Étale groupoids correspond to inverse quantal frames, with Morita equivalence and principal bundles captured by Hilbert bimodules with a Hilbert basis and valued inner products, paralleling C*-algebraic imprimitivity bimodules (Quijano et al., 2018). Morita equivalence is characterized categorically by:

  $$X \otimes_R X^* \cong Q, \quad X^* \otimes_Q X \cong R,$$

  for Q and R inverse quantal frames and X a Hilbert bimodule with Hilbert basis.

• State-Sum and Modular Functor Realizations: The assignment of values to surfaces in 3d topological field theories via block functors and state-sum constructions uses bimodule categories (unitarized as Hilbert bimodules in appropriate settings) to compute extended topological invariants and mapping class group representations (Fuchs et al., 2019).
  • The modular functor factors through "gluing categories" (framed centers of module categories or their unitary analogues) and block spaces are naturally spaces of module natural transformations (or their Hilbert-space enriched analogues).

4. Morita Theory, Finite Index, and Dualizability

• Morita Theory: The classification of 2-categories via projective 2-representations and retracts of principal representations, well-developed for finitary additive 2-categories, generalizes to the unitary analytic context by replacing direct sums/retracts with Hilbert direct sums, projections, and adjoints (Mazorchuk et al., 2013, Chen et al., 7 Oct 2024).
  • Morita equivalence classes are given by examining bicategories up to biequivalence of module 2-categories.
• Finite Index and Approximation Properties: A finite index Hilbert bimodule between algebras A and B (see, e.g., (Forough et al., 2016)) ensures transfer of key analytic properties (WEP, (Q)WEP, LLP, CBAP, (S)OAP, nuclearity, exactness) between A and B, reinforcing the "finite type" intuition. Imprimitivity bimodules, a special case with indices equal to the identity, implement strong Morita equivalence.
  • Non-finite index cases may still retain some transfer properties via amenable crossed products or conditional expectations.
• Rigidity and Separability in Higher Category Terms: Rigid and separable algebras in a monoidal 2-category furnish finite semisimple categories of modules/bimodules; a separable algebra is characterized by a splitting of the adjunction and a nonzero categorical dimension

  $$\mathrm{Dim}_{\mathfrak{C}(A)} := \mathrm{Tr}(\mu) \cdot \mathrm{Tr}((\mu^{-1})^*).$$

  This notion guarantees dualizability and full semisimplicity for the associated 2-category, essential for “unitarizability” and topological field theory applications (Décoppet, 2022).

5. Universal Constructions, Completion, and Multiplier Theory

Several universal properties and completions support the categorical and analytic structure:

• Quasi-Multiplier and Multiplier Bimodules: Every Hilbert C*-bimodule V admits a universal (maximal, strictly essential) quasi-multiplier extension QΜ(V), characterized as the isometric Banach space of bounded (A,B)-bilinear maps, with a topological description via the quasi-strict topology (Pavlov et al., 2010).

  $QΜ(V) = \text{Hom}_{A,B}(A\times B, V)$

  and the closure in this topology gives the universal ambient for possible bimodule extensions, essential for completion phenomena and coherence in higher categorical constructions.

• Multipliers and Crossed Products: In the pro-C*-algebra setting, the multiplier bimodule M(X) = L_A(A, X) and its strict topology serve as the completion for bimodules, and all morphisms extend uniquely to their multipliers (Joiţa et al., 2014).
  • The structure is stable under crossed product operations: $$A\rtimes_X \mathbb{Z} \hookrightarrow M(A)\rtimes_{M(X)} \mathbb{Z} \subset M(A\rtimes_X \mathbb{Z})$$.
  • This universality and multiplicative stability are key when constructing saturated or “unitarized” 2-categories.

6. Algebraic and Quantum Symmetry Aspects

• Tannaka–Krein Equivalence and Quantum Automorphisms: Any concrete unitary 2-category of finite type Hilbert–M–bimodules (with M a direct sum of finite-dimensional matrix algebras) is shown to be equivalent to the category of unitary equivariant corepresentations (on such bimodules) of a certain algebraic quantum group $\mathbb{G}$ acting on M by a canonical coaction (Rollier, 6 May 2024).
  • Matrix coefficients $\#1{H}{\xi}{\eta}$, arising from dense subspaces in bimodules, generate a universal *-algebra $\mathcal{A}$ with a comultiplication determined by category-theoretic tensor product properties.
  • As applied to discrete quantum structures (e.g., Cayley graphs), quantum automorphism groups $\qaut(\Pi)$ are defined so as to preserve adjacency structure, recover (via the 2-category) universal quantum symmetry assigned to the bimodule category generated by the graph (Rollier, 6 May 2024).
• Planar Algebras and Module Classification: For generalized Temperley-Lieb-Jones 2-categories, classification of unitary modules (module *-categories) up to equivalence is achieved via combinatorial data (balanced fair weighted graphs), with precise correspondence to analytic data of the modules (Ferrer et al., 2019).

7. Quantum-Classical Correspondence and State-Sum Constructions

• Classical Limit and Symplectic Dual Pairs: The functorial assignment associating symplectic dual pairs to Hilbert bimodules and vice versa via strict deformation quantization shows that the passage from quantum to classical models preserves the 2-categorical structure (objects, 1-morphisms, and 2-morphisms correspond under classical limits and tensor composition is respected) (Feintzeig et al., 12 Mar 2024).
• State-Sum Modular Functor: The modular functors constructed using bimodule categories as state-sum variables admit a unitarization in which all categories, functors, and natural transformations are carried out in the framework of finite type Hilbert bimodules, giving fully computable extended field theory invariants (Fuchs et al., 2019).

---

This synthesis presents the comprehensive structure and significance of the unitary 2-category of finite type Hilbert bimodules, elucidating its analytic, categorical, and algebraic realizations, universal properties, completion theory, dualizability, and its central role in quantum algebra, operator algebras, and mathematical physics.