Reconstruction Module in Tensor Categories

• Reconstruction Module is a framework that defines methods to recover module categories from internal data in tensor categories using algebraic and categorical tools.
• It employs structures such as quasi-finite coalgebra objects and lax module monads to extend classical reconstruction beyond finite, rigid settings.
• The approach bridges classical algebra-object methods with modern techniques, enabling Morita equivalence and a categorical proof of the Hopf–monadic fundamental theorem.

A reconstruction module, within the context of modern tensor category theory and module category theory, refers to the set of methods, algebraic structures, and categorical formalisms that enable one to recover or "reconstruct" a module category from internal data in the ambient monoidal (or tensor) category. The classical paradigm applies to finite tensor categories and rigid monoidal categories, where reconstruction proceeds via algebra objects. In the infinite or non-rigid settings, the reconstruction module must deploy more general structures—such as quasi-finite coalgebra objects and lax module monads—to effect the reconstruction and classification of module categories. This shift is motivated by the failure of the classical algebra-object-based approach once rigidity and finiteness are relaxed, necessitating deeper examination of categorical projectives, coalgebraic invariants, and equivariant (lax) monad structures.

1. Extension of Reconstruction Theory Beyond the Finite–Rigid Setting

In the classical Etingof–Ostrik framework, a finite tensor category $\mathcal{C}$ and its module category $\mathcal{M}$ admit reconstruction as the category of right modules over an algebra object $A \in \mathcal{C}$: $\mathcal{M} \simeq \mathrm{mod}_{\mathcal{C}} A.$ The new approach replaces $A$ by a quasi-finite coalgebra object when $\mathcal{C}$ and $\mathcal{M}$ are only locally finite, not finite; thus, reconstruction proceeds as

$\mathcal{M} \simeq \mathrm{comod}_{\mathcal{C}}(C)$

where $C$ is a quasi-finite coalgebra object in $\mathcal{C}$ [(Stroiński et al., 1 Sep 2024), Section 1].

This shift is necessary as the direct use of algebra objects becomes untenable outside the finite and rigid environments. The methods require a careful analysis of internal projective and injective objects in $\mathcal{M}$ and their categorical (co)ends, rendering the reconstruction module inherently coalgebraic and suitable for truly infinite settings.

2. Treatment of Non-Rigid Monoidal Categories via Lax Module Monads

When the monoidal category $\mathcal{C}$ lacks rigidity (i.e., not every object admits left and right duals), classical strategies to produce internal algebra or coalgebra objects from $\mathcal{M}$ fail. The reconstruction module in this context is a lax $\mathcal{C}$-module monad, a generalization of the algebra-object: $T_\mathrm{ma}: T(V \odot M) \to V \odot T(M)$ where $T$ is a monad on $\mathcal{M}$, $V \in \mathcal{C}$, $M \in \mathcal{M}$, and $T_\mathrm{ma}$ (the "module action structure map") may fail to be invertible ((Stroiński et al., 1 Sep 2024), Def. 2.4, Sec. 3). When $\mathcal{C}$ is rigid, every lax module monad is automatically strong, and hence equivalent to giving an algebra object in $\mathcal{C}$ ((Stroiński et al., 1 Sep 2024), Prop. 3.16, Prop. 3.21), but when rigidity fails, lax module monads capture the required generalization.

The central reconstruction equivalence in this context states that for a suitable "generator" $X \in \mathcal{M}$,

$\mathcal{M} \simeq EM(\mathrm{Hom}(X, - \odot X))$

where $EM$ is the Eilenberg-Moore category of the lax module monad ((Stroiński et al., 1 Sep 2024), Thms. 4.1, 4.2). The requisite coherence conditions—associativity and unit maps for $T_\mathrm{ma}$—mirror the underlying monoidal structure but allow for non-invertibility due to the absence of duals.

3. Concrete Realization for Comodule Categories and Hopf Trimodule Algebras

For the pivotal case where $\mathcal{C} = \mathrm{com}_{fd}[B]$ is the category of finite-dimensional left comodules over a bialgebra $B$ (not necessarily Hopf), the reconstruction module is explicitly realized as a category of contramodules over a Hopf trimodule algebra $A$ ((Stroiński et al., 1 Sep 2024), Sec. 5).

A Hopf trimodule consists of a $B$-bicomodule equipped with a left $B$-action satisfying suitable compatibility diagrams. If the action functor in $\mathcal{M}$ has both left and right adjoints, there exists such a Hopf trimodule algebra $A$ with an equivalence: $$EM(\mathrm{Hom}(X, - \odot X)) \simeq A\text{-Contramod}.$$ When $B$ is Hopf, this recovers the established tensor-categorical reconstruction theorem for finite tensor categories. The authors provide explicit computations (e.g., (Stroiński et al., 1 Sep 2024), Ex. 5.12), contrasting the classical algebra-object based methods and the Hopf trimodule approach.

4. Morita and Eilenberg–Watts-Type Characterizations

The reconstruction module incorporates a bicategorical perspective, relating module category equivalences to Morita equivalence classes of module monads. A right exact lax $\mathcal{C}$-module functor between Eilenberg-Moore categories $EM(T) \rightarrow EM(S)$ corresponds to a "biact functor" (or bimodule functor) in the 2-category of right exact lax $\mathcal{C}$-module endofunctors ((Stroiński et al., 1 Sep 2024), Thm. 4.18).

Given two such monads $S$ and $T$, Morita equivalence is witnessed by the existence of $S$–$T$ and $T$–$S$ biact functors whose respective compositions (via relative tensor product) yield the identity functors up to isomorphism ((Stroiński et al., 1 Sep 2024), Def. 4.20, Prop. 4.21). The equivalence between module categories is then classified by the Morita classes of right exact lax $\mathcal{C}$-module monads ((Stroiński et al., 1 Sep 2024), Thm. 4.24).

5. Fusion Operators, Categorical Proofs, and the Hopf–Monadic Fundamental Theorem

In the context of Hopf monads (as in Bruguières, Lack, and Virelizier), a fusion operator $H: T \circ (V \odot -) \to V \odot T(-)$ is central. The paper demonstrates that such a fusion operator is categorically a coherence cell for the module functor structure on $T$ ((Stroiński et al., 1 Sep 2024), e.g., following Prop. 4.12), so the property of having a unique invertible cell characterizes Hopf monads categorically.

This enables a categorical proof—bypassing explicit algebraic computations—of a variant of the fundamental theorem of Hopf modules: the Eilenberg-Moore category of a Hopf monad, equipped with its canonical extended module structure, is equivalent to the module category of the associated Hopf trimodule algebra. That is,

$EM(T) \simeq \mathrm{Mod}_{A}$

whenever the appropriate fusion (coherence) conditions are satisfied ((Stroiński et al., 1 Sep 2024), Section 4), reproducing the Hopf–monadic fundamental theorem in this broad context.

6. Bridging Classical and Infinite/Non-Rigid Settings

The comprehensive approach developed in (Stroiński et al., 1 Sep 2024) establishes a bridge between earlier finite, rigid module-category reconstruction (via algebra objects) and the more general infinite and non-rigid settings (via quasi-finite coalgebra objects and lax module monads). The methodology is robust to the loss of finiteness and rigidity, incorporating diagrammatic arguments (notably Linton coequalizers), generalized Morita theory, and the use of categorical projectives/injectives.

These results provide an extensible foundation for further paper in tensor-categorical representation theory, quantum groups, vertex operator algebras, and related domains where module categories over general monoidal categories are central objects of paper. The categorical perspective adopted—especially regarding fusion operator coherence and Morita classification—offers a unified and flexible toolkit for the analysis and reconstruction of module categories in a variety of settings.