Morita 2-Category of Super Vector Spaces

• The Morita 2-category of super vector spaces is a higher-categorical framework that classifies superalgebraic structures using ℤ₂-graded rings and parity reversal functors.
• It leverages super Morita equivalence where progenerator bimodules and dualities establish isomorphisms between module categories over super rings and algebras.
• Applications span superrepresentation theory, superalgebraic geometry, and topological field theories via fusion 2-categories and explicit constructions like the Clifford/Fock module.

The Morita 2-category of super vector spaces is a higher-categorical structure designed to classify and relate categories of supermodules, superalgebras, and their bimodules in a symmetric monoidal setting enriched with parity (ℤ₂-grading). The foundational principles are sourced from super Morita theory (Kwok, 2013), higher Morita-theoretic results in fusion 2-categories (Décoppet, 2022, Décoppet, 2022), and their interplay with representation, duality, and projective 2-representations (Fiorenza et al., 20 Sep 2025). This structure underpins the categorical symmetry in superrepresentation theory, superalgebraic geometry, and related aspects of topological field theories.

1. Fundamental Structures: Super Rings, Modules, and the Parity Functor

A super ring $R$ is a ℤ₂-graded ring, $R = R_0 \oplus R_1$, with multiplication respecting the grading (i.e., $|ab| = |a| + |b| \pmod{2}$), and sign factors appear as required for homogeneous elements. Supermodules over $R$ inherit ℤ₂-gradings; the $R$-action is required to respect the grading.

Parity reversal, $\Pi$, is a fundamental functor: for a supermodule $M$, the shifted module $\Pi M$ has $(\Pi M)_0 = M_1$ and $(\Pi M)_1 = M_0$, with module actions twisted by appropriate signs. This functor is essential for constructing generators, progenerators, and handling duality in super Morita theory.

The distinction between categorical Hom (parity-preserving maps) and internal Hom (all degree-homogeneous maps) is critical in the super context: the latter underlies the definition of generators and, ultimately, equivalence of supermodule categories.

2. Super Morita Context and Equivalences

Given a right $R$-module $P$, the super Morita context comprises the tuple $(R, P, Q, S; \alpha, \beta)$:

• $Q = P^* = \operatorname{Hom}_R(P, R)$ is the dual module, graded appropriately.
• $S = \operatorname{End}_R(P)$ is the super ring of $R$-linear endomorphisms, again graded.

The canonical pairings are:

• $\alpha: Q \otimes_S P \to R$, with $\alpha(q \otimes p) = q(p)$,
• $\beta: P \otimes_R Q \to S$, with $\beta(p \otimes q) = (x \mapsto p \cdot q(x))$.

If $P$ is a progenerator, $\alpha$ and $\beta$ are isomorphisms of bimodules in the super setting; these isomorphisms establish the equivalence of the categories of supermodules over $R$ and $S$. The functors $-\otimes_R Q: \mathfrak{M}_R \to \mathfrak{M}_S$ and $-\otimes_S P: \mathfrak{M}_S \to \mathfrak{M}_R$ are mutually inverse equivalences.

Super Morita equivalence generalizes classical Morita theory, with explicit attention to ℤ₂-grading, parity shift, and sign factors in all constructions.

3. 2-Categorical and Bicategorical Formulation

The Morita 2-category of super vector spaces, denoted typically as $2\,\mathrm{sVect}_k$, admits the following stratification:

• Objects: Finite-dimensional super-commutative $k$-algebras (or, more generally, superalgebras and superrings).
• 1-morphisms: Progenerator bimodules—i.e., bimodules over $(S, R)$ which are both projective and generators, plus their parity shifts.
• 2-morphisms: Intertwiners, i.e., bimodule maps respecting parity.

Composition of 1-morphisms is super tensor product; the passage from algebra bundles, gerbes, and associated modules (as in 2-vector bundles (Kristel et al., 2021)) is handled in this bicategorical language.

Results from super Morita theory (Kwok, 2013) directly inform this structure: two superalgebras are Morita equivalent if and only if their supermodule categories are equivalent, with progenerator bimodules mediating these equivalences. The classification of 1-morphisms via tensor products is categorical, with invertibility governed by the underlying algebraic properties (e.g., invertible Clifford algebras (Fiorenza et al., 20 Sep 2025)).

Semisimplicity and dualizability are inherited from the bicategory of algebras, bimodules, and intertwiners. The 2-category $\mathrm{KV}$ of Kapranov-Voevodsky vector spaces (Lorand et al., 2019) and its super-analogue provide a template for handling representation bifunctors and strict duality involutions.

4. Super Azumaya Algebras and Explicit Examples

Super Azumaya algebras serve as model objects for Morita equivalences in the super category. Given a commutative superalgebra $R$, $A$ is a super Azumaya algebra if it is:

1. Faithfully linear and finitely generated projective as an $R$-module,
2. The canonical map $$\phi: A \otimes_R A^{op} \to \underline{\operatorname{End}}_R(A)$$ is an isomorphism of $R$-superalgebras,
3. Its supercenter consists of elements $x$ with $ax = (-1)^{|a||x|} xa$ for all homogeneous $a \in A$.

The super skew field $\mathbb{D} = k[\theta], \theta^2 = -1$ is central simple and provides a canonical example. The equivalence of module categories between $R$ and $A$-modules is realized using the supercommutant of an $(A,A)$-bimodule $$M^A = \{m \in M: a m = (-1)^{|a||m|} m a,\, \forall\, a \}$$, with the explicit isomorphism $$\underline{\operatorname{Hom}}_{A^e}(A, M) \cong M^A$$.

This concrete machinery connects superalgebraic geometry and representation theory to Morita equivalence; it is explicit enough to classify equivalence classes via the intermediary of bimodules and their parity structures.

5. Projective 2-Representations and Freeness Phenomena

The freeness property established for invertible projective 2-representations (Fiorenza et al., 20 Sep 2025) asserts that, given a symmetric monoidal 2-category $2\mathcal{V}$ (e.g., $2\,\mathrm{sVect}_k$) with duals, assignments of invertible objects and 1-morphisms automatically yield a projective 2-representation equipped with a canonical 2-cocycle in $\mathrm{Pic}(\mathcal{V})$: $$l_{\Xi_{ijk}} = \mathrm{tr}(M_{f_{ik}}^{-1} \circ M_{f_{jk}} \circ M_{f_{ij}})$$ where the trace is taken internally in the 2-category.

The Clifford/Fock construction is paradigmatic:

• To an anti-involutive Hilbert space $(H, \alpha)$, assign the Clifford algebra $\mathrm{Cl}(H, \alpha)$ (an invertible superalgebra).
• To a Lagrangian correspondence $L$, assign the Fock module $F(L) = \wedge^\bullet(\alpha(L))$, which is a super-bimodule between Clifford algebras associated to $H_1$ and $H_2$.

Associativity and coherence constraints are governed by Pfaffian lines, which appear as 2-cocycles—precisely the measure of projectivity in such 2-representations. The full machinery is essential to representation theory of higher-categorical objects and topological field theories with defects.

6. Extensions to Fusion 2-Categories and Drinfeld Centers

Morita theory in fusion 2-categories (Décoppet, 2022, Décoppet, 2022) subsumes the super scenario. In this framework:

• Separable (rigid) algebras in compact semisimple tensor 2-categories (including $2\,\mathrm{sVect}_k$) correspond to Morita objects.
• Morita equivalence is characterized by equivalence of the full 3-categories of separable module 2-categories.
• The Drinfeld center $\mathcal{Z}(\mathcal{C})$ is invariant under Morita equivalence and is always a finite semisimple 2-category.

A strongly fusion 2-category is one whose monoidal unit has endomorphisms equivalent to $\mathbf{Vect}$ (bosonic) or $\mathbf{SVect}$ (fermionic, i.e., the super case). Every fusion 2-category is Morita equivalent to a tensor product of a strongly fusion 2-category and an invertible one.

Dimension formulas such as $$\mathrm{Dim}(Mod(\mathcal{B})) = 1/\mathrm{dim}(\mathcal{B})$$ categorize underlying structures, while separability implies full 4-dualizability (important for extended TQFTs).

7. Special Cases: Characteristic 2 and Infinite-Dimensional Constructions

In characteristic 2, ordinary super vector space notions are replaced by the category $\mathrm{sVec}_2$ (Kaufer, 2018), where objects are vector spaces with a differential $d$ satisfying $d^2=0$, morphisms are linear maps commuting with $d$, and commutativity of multiplication is twisted via $ab = ba + d(b)d(a)$. d-algebras and their module categories fill the same role as superalgebras in characteristics $\neq 2$.

In infinite-dimensional contexts (e.g., 2-vector bundles or stringor representations (Kristel et al., 2021, Kristel et al., 2023)), objects are modeled as C*-algebras, von Neumann algebras, or gerbes, with bimodules and intertwiners forming the 1- and 2-morphisms; actions of higher groups (e.g., the string 2-group) are accommodated within this Morita 2-category by explicit representations on hyperfinite factors.

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The Morita 2-category of super vector spaces provides a precise framework for classification of supermodule categories up to equivalence, explicit construction of categorical invariants, projective 2-representations, and duality structures foundational to supergeometry, higher representation theory, and topological field theory. Its architecture reflects the interplay of superalgebraic and categorical data, leveraging ℤ₂-graded structures, parity functors, progenerator modules, and their tensorial composition as the basis for symmetry and equivalence in modern mathematics.