Exact Module Category in Tensor Categories

• Exact module category is a finite abelian, k-linear category with an exact action from a finite tensor category, preserving projectivity and enabling coherent duality.
• It encompasses exact sequences where tensor functors and Frobenius–Perron dimensions generalize classical Hopf algebra sequences within a categorical framework.
• This structure underpins robust extension theory and Morita equivalence, with applications in modular categories, topological field theories, and modern representation theory.

An exact module category is a categorical structure that generalizes the classical notion of exactness from module theory to the context of finite tensor categories and their module categories. In such settings, an exact module category enables robust extension theory and dualities, is essential for describing Morita theory of tensor categories, and provides a necessary foundation for the theory of extensions and exact sequences of tensor categories.

1. Definition and Formulations of Exact Module Categories

In the setting of finite tensor categories, an exact module category is defined as a finite abelian, $k$-linear category $\mathcal{M}$ equipped with an exact action of a finite tensor category $\mathcal{C}$—that is, the action $$\otimes : \mathcal{C} \times \mathcal{M} \to \mathcal{M}$$ is $k$-bilinear and right exact in each variable, and for every projective object $P \in \mathcal{C}$ and every $M \in \mathcal{M}$, the object $P \otimes M$ is projective in $\mathcal{M}$ (Yadav, 2023).

This notion extends the classical paradigm where exactness in a module category over a ring means that projectives remain projective under base change. The formalism allows for the development of extension, lifting, duality, and module-theoretic operations in the categorical regime. In many influential works, including those of Etingof, Ostrik, and their collaborators, exact module categories are those for which the natural module action preserves enough projectivity and exactness to allow for the formation of well-behaved dual categories and for the implementation of Morita theory (Meir et al., 2010).

2. Exact Sequences with Respect to a Module Category

The notion of an exact sequence of tensor categories "with respect to a module category" generalizes classical exact sequences of Hopf algebras and their representation categories (Etingof et al., 2015). Instead of requiring the kernel to have a tensor functor to $\mathrm{Vec}$ (i.e., to be Tannakian), the setup replaces $\mathrm{Vec}$ by $\mathrm{End}(\mathcal{M})$, the finite tensor category of right-exact endofunctors of an indecomposable exact left $A$-module category $\mathcal{M}$. An exact sequence is then a diagram of tensor functors: $$A \longrightarrow B \overset{F}{\longrightarrow} C \boxtimes \mathrm{End}(\mathcal{M})$$ subject to certain normality and surjectivity conditions:

• For every $X \in B$, there exists a subobject $X_0 \subset X$ so that $F(X_0)$ is the largest subobject of $F(X)$ lying in $\mathrm{End}(\mathcal{M})$ (or, dually, there exists a quotient $X_0$ of $X$ such that $F(X_0)$ is the largest quotient of $F(X)$ in $\mathrm{End}(\mathcal{M})$).
• $F$ is surjective and its tensor subcategory kernel is $A$.

These formulations are shown to be equivalent and lead to recovery of classical dimension formulas such as

$$\mathrm{FPdim}(B) = \mathrm{FPdim}(A)\cdot\mathrm{FPdim}(C),$$

where $\mathrm{FPdim}(-)$ denotes Frobenius–Perron dimension (Etingof et al., 2015).

3. Examples: Deligne Tensor Product and Duality

A fundamental example of an exact sequence in this sense is given by the Deligne tensor product of finite tensor categories: for $B = C \boxtimes A$, the functor to $C \boxtimes \mathrm{End}(\mathcal{M})$ with $A$ acting via its natural action on $\mathcal{M}$ yields an exact sequence. This example also demonstrates that the theory generalizes the sequence of categories of comodules arising from an exact sequence of finite-dimensional Hopf algebras, even when $A$ does not possess a fiber functor (Etingof et al., 2015).

A central structural result is that these exact sequences are self-dual. Dualizing an exact sequence with respect to an indecomposable exact $C$-module category $\mathcal{N}$ produces another exact sequence: $$A^*_\mathcal{M} \longrightarrow \mathrm{End}(\mathcal{N}) \boxtimes B^* \longrightarrow C^*_\mathcal{N},$$ where duality involves the module categories as "base points." The dual sequence maintains the inclusion of $C$ in the kernel, and the properties of surjectivity and normality are reflected on the dual side. Taking the double dual recovers the original sequence, mimicking the behavior in Hopf algebra theory (Etingof et al., 2015).

4. Semisimplicity and Extension Theory

When $A$ and $C$ are fusion categories (i.e., semisimple), and $\mathcal{M}$ is a semisimple module category, any exact sequence of the form above forces the middle term $B$ to also be a fusion category. The argument utilizes an equivalence of module categories: $$B \boxtimes_A \mathcal{M} \simeq C \boxtimes \mathcal{M},$$ which, given the semisimplicity of $C$ and $\mathcal{M}$, implies the semisimplicity of $B \boxtimes_A \mathcal{M}$, and thus of $B$ itself. This demonstrates stability of semisimplicity under the passage through exact sequences in this enhanced categorical context (Etingof et al., 2015).

5. Relation to Hopf Algebras and Broader Framework

The categorical notion of exact sequence with respect to a module category generalizes the theory of exact sequences for tensor categories arising from Hopf algebras, as first treated by Bruguières and Natale. In the Hopf algebraic setting, for a short exact sequence $A \hookrightarrow B \twoheadrightarrow C$ of finite-dimensional Hopf algebras, one obtains an exact sequence of their representation categories. The categorical framework replaces the role of Tannakian subcategories with that of endofunctor categories $\mathrm{End}(\mathcal{M})$, allowing the treatment of Deligne products and other constructions not accessible via Tannakian formalism alone. In this expanded setting, foundational properties such as duality of exact sequences and Frobenius–Perron dimension formulae continue to hold (Etingof et al., 2015).

6. Implications and Applications

The theory of exact module categories and exact sequences with respect to module categories underpins a robust extension and duality theory in finite tensor categories. It connects categorical notions to classical algebraic concepts and enables systematic handling of extensions, dualities, and module-theoretic properties in modern categorical representation theory. The structure described here enables the description and classification of extensions of tensor categories, provides tools for computation and comparison of module categories, and ultimately situates module categories as central agents in the paper of tensor category invariants and Morita equivalence.

When the endpoint categories are fusion categories, or more generally semisimple, semisimplicity automatically extends to the entire sequence, ensuring that "nice" properties persist. This theoretical framework naturally encompasses and generalizes the theory of exact sequences of Hopf algebras and opens the way for further developments in the structure theory of tensor categories, including their applications in modular categories, topological field theories, and beyond.

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References

• (Etingof et al., 2015)
• (Meir et al., 2010)
• (Yadav, 2023)