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arxiv: 2604.10753 · v1 · submitted 2026-04-12 · 🧮 math.QA · math.CT· math.RT

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Finite Pre-Tensor Categories that are Morita Equivalent to Finite Tensor Categories

Thibault D. D\'ecoppet , Mateusz Stroi\'nski
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Pith reviewed 2026-05-10 15:17 UTC · model grok-4.3

classification 🧮 math.QA math.CTmath.RT
keywords finite pre-tensor categoriesMorita equivalenceDrinfeld centerfinite tensor categoriesbimodulesright exact tensor productabelian categoriesdual objects
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The pith

A finite pre-tensor category is Morita equivalent to a finite tensor category precisely when its Drinfeld center is a finite tensor category.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a complete characterization of finite pre-tensor categories that stand in Morita equivalence to finite tensor categories. It shows this equivalence holds exactly when the Drinfeld center of the pre-tensor category itself forms a finite tensor category. This matters because finite pre-tensor categories generalize finite tensor categories by requiring duals only for projective objects rather than all objects, and the bimodule construction already produces many examples of the former from the latter. A reader who accepts the result gains a practical test for when these generalized categories belong to the same Morita class as ordinary tensor categories.

Core claim

A finite pre-tensor category C is Morita equivalent to a finite tensor category if and only if the Drinfeld center of C is a finite tensor category. The authors extend the standard Morita theory for tensor categories to the pre-tensor setting by using right-exact tensor products and bimodules, then apply the Drinfeld center to obtain the if-and-only-if statement. They further note higher algebraic consequences that follow from the characterization.

What carries the argument

The Drinfeld center of a finite pre-tensor category, which serves as the invariant that detects whether the category is Morita equivalent to one in which every object possesses a dual.

If this is right

  • Every category of bimodules over an algebra in a finite tensor category can be tested for Morita equivalence to a tensor category simply by computing its Drinfeld center.
  • The Morita classes of finite tensor categories sit inside the larger Morita classes of finite pre-tensor categories as a distinguished subclass identified by the center condition.
  • Higher algebraic invariants built from the Drinfeld center, such as modular data or Frobenius-Perron dimensions, become available exactly for those pre-tensor categories that meet the criterion.
  • Classification problems for finite tensor categories can be enlarged to pre-tensor categories while preserving the Morita equivalence relation via the center test.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result supplies a criterion that could be used to decide whether a given abelian category with right-exact tensor product can be realized inside a tensor category up to Morita equivalence.
  • If the center condition fails, the pre-tensor category may still carry useful structure but cannot be replaced by an ordinary tensor category without changing its Morita class.
  • The characterization suggests that attempts to classify all finite pre-tensor categories can be reduced, in the Morita-equivalent-to-tensor case, to the already-studied classification of finite tensor categories.

Load-bearing premise

The Drinfeld center construction, when applied to a finite pre-tensor category defined via right-exact tensor product and duals on projectives, yields a tensor category precisely when the original category satisfies the stated Morita equivalence.

What would settle it

Exhibit a single finite pre-tensor category whose Drinfeld center is a finite tensor category yet which cannot be realized as the category of bimodules over any algebra in a finite tensor category.

read the original abstract

A finite pre-tensor category is a finite abelian category equipped with a right exact tensor product for which every projective object has duals. Finite tensor categories, for which every object has duals, are notable examples. More generally, the category of bimodules over an algebra in a finite tensor category is a finite pre-tensor category. In particular, it is natural to extend the notion of Morita equivalence between finite tensor categories to finite pre-tensor categories. We characterize completely those finite pre-tensor categories that are Morita equivalent to finite tensor categories. More precisely, we show that a finite pre-tensor category $\mathcal{C}$ is Morita equivalent to a finite tensor category if and only if the Drinfeld center of $\mathcal{C}$ is a finite tensor category. We also discuss higher algebraic consequences of our characterization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper defines finite pre-tensor categories as finite abelian categories equipped with a right-exact tensor product in which every projective object has a dual. It extends the notion of Morita equivalence to this setting via bimodule categories equipped with right-exact tensor products. The central result is a complete characterization: a finite pre-tensor category C is Morita equivalent to a finite tensor category if and only if the Drinfeld center Z(C) is itself a finite tensor category (i.e., rigid). The manuscript also discusses higher algebraic consequences of this characterization.

Significance. If the result holds, the characterization supplies a concrete, checkable criterion (rigidity of the Drinfeld center) for membership in the Morita class of a finite tensor category. This extends classical Morita theory for tensor categories in a natural way and may streamline arguments in the study of module categories and higher categorical structures. The paper's use of standard constructions (Drinfeld center, bimodules) without introducing new ad-hoc parameters is a strength.

minor comments (2)
  1. The abstract and introduction could more explicitly preview the higher algebraic consequences mentioned, perhaps with a brief pointer to the relevant later section, to help readers assess the broader impact immediately.
  2. Notation for the extended Morita equivalence (via bimodules with right-exact tensor products) should be introduced with a short comparison table or sentence contrasting it to the usual tensor-category Morita equivalence to reduce potential confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly identifies the central characterization and its potential utility in extending Morita theory. We will incorporate minor improvements to clarity and exposition in the revised version.

Circularity Check

0 steps flagged

No circularity: standard characterization via Drinfeld center on extended Morita theory

full rationale

The paper defines finite pre-tensor categories (finite abelian with right-exact tensor product, projectives dualizable) and extends Morita equivalence via bimodules with right-exact tensor products. It then states the theorem that C is Morita equivalent to a finite tensor category precisely when Z(C) is itself a finite tensor category. This is a non-trivial iff claim whose proof must verify that the coend/half-braiding formulas for the center produce full duals exactly under the Morita condition; nothing in the provided abstract or description reduces the statement to a definition, a fitted parameter renamed as prediction, or a self-citation chain. The construction is self-contained against external categorical benchmarks and does not invoke load-bearing prior results by the same authors that would close the loop.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on standard definitions and constructions in category theory without new free parameters or invented entities.

axioms (3)
  • domain assumption Finite abelian categories equipped with a right exact tensor product in which every projective object has a dual form a finite pre-tensor category.
    This is the explicit definition given in the abstract.
  • domain assumption The Drinfeld center of a finite pre-tensor category can be formed and checked for the property of being a finite tensor category.
    Standard construction invoked in the characterization.
  • domain assumption Morita equivalence extends naturally from finite tensor categories to finite pre-tensor categories via bimodule categories.
    The abstract states that the notion is extended in this way.

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discussion (0)

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