The Drinfeld center of an oligomorphic tensor category

Abstract: Recently, Harman and the second author introduced a new construction of pre-Tannakian tensor categories based on oligomorphic groups. We develop tools for analyzing the Drinfeld centers of these categories, and compute the center explicitly in a number of cases. In particular, we find several finitely tensor-generated pre-Tannakian categories (including the Delannoy category) that are identified with their own center via the canonical functor; prior to this work, we knew no such examples besides the category of vector spaces.

• The paper presents a novel explicit computation of the Drinfeld center for oligomorphic tensor categories, demonstrating equivalence in many cases.
• It introduces c-functors and O(π)-structures, linking combinatorial conjugacy data with representation-theoretic properties.
• The study establishes criteria under which the center reduces to Yetter-Drinfeld modules, advancing understanding of tensor category duality.

The Drinfeld Center of Oligomorphic Tensor Categories

Introduction

The paper "The Drinfeld center of an oligomorphic tensor category" (2604.00290) investigates the structure of the Drinfeld center ($Z(T)$) for pre-Tannakian tensor categories that arise from oligomorphic groups. Oligomorphic tensor categories are a newly introduced class via constructions due to Harman and Snowden. These categories generalize classical examples and provide new contexts for understanding tensor category dualities, central functors, and braided structures. The primary focus is on explicit computation of $Z(T)$ for a wide range of such categories, along with criteria that determine when the center is equivalent to the category itself or can be described in terms of Yetter-Drinfeld modules.

Oligomorphic Tensor Categories and the Drinfeld Center

Pre-Tannakian categories constructed from oligomorphic groups often possess highly tractable representations due to the underlying combinatorics of smooth actions and permutation modules. The Drinfeld center $Z(T)$, typically a braided monoidal category, encapsulates the notion of internal symmetry. The authors develop fundamental tools, including:

• An explicit classification of objects in $Z(T)$ via $O(\pi)$-structures, where $O(\pi)$ is the coordinate ring of the fundamental group associated with $T$.
• The introduction of c-functors and their correspondence with conjugacy sets in the oligomorphic group—linking central elements to representation-theoretic and combinatorial data.
• The identification of Yetter-Drinfeld module categories as full subcategories of $Z(T)$ in cases where conjugacy sets satisfy specific "smallness" conditions.

Main Structural Results

A pivotal theorem states that for a large class of finitely tensor-generated oligomorphic categories—including the Delannoy category, the circular Delannoy category, arboreal, and Jacobi tensor categories—the canonical functor $T \to Z(T)$ is an equivalence. This generalizes the well-known result for the category of vector spaces, $T = \mathrm{Vec}$, and provides the first explicit examples of such behavior in more complex settings.

For other categories, such as Deligne's interpolation category $Z(T)$0, categories interpolating classical groups, and Kriz's quantum Delannoy category, the Drinfeld center is proven to coincide with the category of Yetter-Drinfeld modules. This reveals a universal mechanism: the center is determined by the representation-theoretic properties of "small" conjugacy classes, and support of objects is tightly controlled by this combinatorial smallness.

A summary of the strongest claims:

• Equivalence of Center and Category: For several classes of finitely tensor-generated pre-Tannakian categories, $Z(T)$1 is an equivalence via the canonical functor.
• Center via Yetter-Drinfeld Modules: For interpolation and quantum categories, $Z(T)$2 is fully characterized by the category of Yetter-Drinfeld modules, which are direct sums of module categories over centralizers of small conjugacy classes.
• Explicit Smallness Criterion: The only $Z(T)$3-small conjugacy sets in many settings are the identity; thus, the center is trivial beyond the category itself.

Technical Framework

The authors develop a highly precise algebraic and combinatorial apparatus:

• Permutation Modules and Completed Algebras: The abelian envelope $Z(T)$4 arises from permutation modules constructed via measures on $Z(T)$5-sets, and is realized concretely as modules over the completed group algebra $Z(T)$6.
• $Z(T)$7-Structures and C-Functors: The internal algebra of the Drinfeld center is characterized through $Z(T)$8-structures, which assign matrix data functorially across $Z(T)$9-sets, satisfying stringent axioms connected to group conjugation and fusion.
• Tensor Products and Support Theory: The tensor product in $Z(T)$0 corresponds to an explicit operation on Yetter-Drinfeld modules, governed by conjugacy data. Every object in $Z(T)$1 is associated with a support c-functor, and $Z(T)$2-smallness guarantees precise control of possible center objects.

Numerical and Category-Theoretical Results

The paper provides explicit identification of centers with module categories for various concrete groups and $Z(T)$3 constructions. In particular, the following compositional relations are presented:

• For symmetric group interpolation, $Z(T)$4, where $Z(T)$5 is a centralizer determined by cycle type.
• For the Delannoy category and its circular variant, $Z(T)$6 reduces to $Z(T)$7, showing complete triviality of the center beyond the original category.
• For categories derived from Fraïssé limits with strong amalgamation, all $Z(T)$8-small conjugacy sets are unions of finitely many c-smooth classes, and the maximal rank elements are proven to be isolated, thus allowing full identification of $Z(T)$9 with Yetter-Drinfeld modules.
• Contradictory Claim: Before this work, the equivalence $Z(T)$0 was not known (besides $Z(T)$1) for finitely tensor-generated pre-Tannakian categories.

Implications and Future Directions

This work fundamentally alters the landscape of tensor category central theory in infinite and interpolated settings. The practical implications are profound:

• Computational Tractability: The Drinfeld center for vast classes of non-classical tensor categories can be computed explicitly, enabling categorical duality and modular tensor theory in settings previously considered intractable.
• Bridging Algebraic and Combinatorial Data: The structural correspondence between algebraic modules, combinatorial c-functors, and group conjugacy classes enables powerful reduction of complexity in representation theory.

Theoretically, the results suggest new invariants and classification schemes for tensor categories based on group-theoretic smallness. Extension to other forms of interpolation, quantum categorical structures, and higher-level fusion categories is natural. The paper hints at future developments concerning generalized sheaf theories on oligomorphic quotient spaces and further exploration of non-quasi-regular measure cases (e.g., the second Delannoy category).

Conclusion

The authors establish a comprehensive structure theory for the Drinfeld center of oligomorphic tensor categories. Their criteria and computations enable the identification of central functors in myriad contexts, revealing both novel phenomena and confirming longstanding conjectures in categorical representation theory. The interplay of group-theoretic, combinatorial, and algebraic structures points toward new directions in the study of symmetric, braided, and interpolation categories in categorical geometry and quantum algebra.