arxiv: 2604.00837 · v2 · submitted 2026-04-01 · 🧮 math.QA · math.CT· math.KT

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· Lean Theorem

Deformations of mixed associators in module categories

Matthieu Faitg , Azat M. Gainutdinov , Christoph Schweigert , Jan-Ole Willprecht

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keywords mixed associatorsmodule categoriesdeformationsDrinfeld centerDavydov-Yetter cohomologyrelative Ext groupsadjoint algebraOcneanu rigidity

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The pith

The cohomology of deformations of mixed associators in module categories over monoidal categories is isomorphic to relative Ext groups in the Drinfeld center.

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

The paper constructs a cochain complex that governs how mixed associators can deform in a module category. It identifies this complex with the Davydov-Yetter complex of the representation functor. For finite categories this identification yields an isomorphism to relative Ext groups involving the unit object and the adjoint algebra of the module category in the Drinfeld center. The result provides explicit dimension formulas for the cohomology groups and proves vanishing in positive degrees for the category itself. It also establishes a generalized rigidity theorem for monoidal functors with coefficients and applies it to examples from Hopf algebras.

Core claim

We set up a cochain complex C^•_mix(M) whose cohomology controls deformations of the mixed associator of a module category M over a k-linear monoidal category C. We show that C^•_mix(M) is isomorphic to the Davydov-Yetter complex of the representation functor ρ : C → End(M). Using our previous results on DY cohomology, we prove that if C and M are finite then the cohomology H^•_mix(M) is isomorphic to the relative Ext groups Ext^•_{Z(C),C}(1, A_M) for the usual adjunction between the Drinfeld center Z(C) and C, where A_M is the adjoint algebra of M. This allows us to give a dimension formula for H^n_mix(M) in terms of certain Hom spaces in Z(C), and also to prove that H^{>0}_mix(C) = 0. We也也

What carries the argument

The mixed cochain complex C^•_mix(M) identified with the Davydov-Yetter complex of the representation functor from C to the endofunctors of M, reducing the deformation problem to relative Ext computations.

Load-bearing premise

The isomorphism between the mixed cochain complex and the Davydov-Yetter complex of the representation functor, which is used without full re-derivation here.

What would settle it

Compute both the mixed cohomology directly for a small finite module category and the corresponding relative Ext groups in the Drinfeld center, and check if their dimensions match.

Figures

Figures reproduced from arXiv: 2604.00837 by Azat M. Gainutdinov, Christoph Schweigert, Jan-Ole Willprecht, Matthieu Faitg.

read the original abstract

We set up a cochain complex $C^\bullet_{\mathrm{mix}}(\mathcal{M})$ whose cohomology controls deformations of the mixed associator of a module category $\mathcal{M}$ over a $\Bbbk$-linear monoidal category $\mathcal{C}$. We show that $C^\bullet_{\mathrm{mix}}(\mathcal{M})$ is isomorphic to the Davydov-Yetter (DY) complex of the representation functor $\rho : \mathcal{C} \to \mathrm{End}(\mathcal{M})$. Using our previous results on DY cohomology (arXiv:2411.19111), we prove that if $\mathcal{C}$ and $\mathcal{M}$ are finite then the cohomology $H^\bullet_{\mathrm{mix}}(\mathcal{M})$ is isomorphic to the relative Ext groups $\mathrm{Ext}^\bullet_{\mathcal{Z}(\mathcal{C}),\mathcal{C}}(\boldsymbol{1},\mathcal{A}_{\mathcal{M}})$ for the usual adjunction between the Drinfeld center $\mathcal{Z}(\mathcal{C})$ and $\mathcal{C}$, where $\mathcal{A}_{\mathcal{M}}$ is the so-called adjoint algebra of $\mathcal{M}$. This allows us to give a dimension formula for $H^n_{\mathrm{mix}}(\mathcal{M})$ in terms of certain Hom spaces in $\mathcal{Z}(\mathcal{C})$, and also to prove that $H^{>0}_{\mathrm{mix}}(\mathcal{C}) = 0$. We also show that the algebra $\mathcal{A}_{\mathcal{M}}$ is the ``full center'' of an algebra in $\mathcal{C}$ realizing $\mathcal{M}$. We furthermore establish a generalized version of Ocneanu rigidity for monoidal functors with coefficients, and provide its application to general (non-exact and non-finite) $\mathcal{C}$-module categories over a fusion category $\mathcal{C}$ such that $\dim(\mathcal{C}) \neq 0$. We spell out these results for module categories defined by finite-dimensional comodule algebras over finite-dimensional Hopf algebras. Examples based on comodule algebras over Sweedler's Hopf algebra are worked out in detail and yield new continuous families of inequivalent non-exact module categories.

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.

Desk Editor's Note private letter to a colleague

The paper defines a mixed cochain complex for mixed associator deformations, identifies its cohomology with relative Ext groups via DY cohomology, and gives new continuous families of non-exact module categories over Sweedler's Hopf algebra.

read the letter

The key takeaway is that the authors define a mixed cochain complex whose cohomology governs deformations of the mixed associator in a module category, prove it matches the Davydov-Yetter complex of the representation functor, and then apply their earlier results to identify the cohomology with relative Ext groups in the Drinfeld center when the category and module are finite. They also produce new families of non-exact module categories from comodule algebras over Sweedler's Hopf algebra. On the positive side, the construction of the mixed complex from the pentagonator and module-associator data is a direct way to capture the deformations. The dimension formula for the cohomology groups in terms of Hom spaces in the center is a practical output. The generalized Ocneanu rigidity result for monoidal functors with coefficients extends the theory to non-exact settings, which is a step forward for broader classification problems. Working out the Sweedler examples in detail shows that these deformations can give continuous families of inequivalent module categories, something that adds concrete value beyond abstract theorems. The softer part is the reliance on the identification with the DY complex. The paper states that C^•_mix(M) is isomorphic to it, but the stress-test note points out that the explicit chain map construction isn't visible in the summary, and any mismatch in how the differentials are defined from the coherence structures could affect the later steps. Since the Ext isomorphism and dimension formula follow from that, a referee would want to see the details of that equivalence to confirm there are no hidden normalization issues. The dependence on the prior arXiv paper for the DY cohomology is acceptable, but it does mean the current manuscript isn't standalone for the full chain of arguments. This work is aimed at people already working in the area of tensor categories, module categories over Hopf algebras, and deformation theory. A reader who needs new invariants for deformations or wants to see how rigidity fails in non-exact cases will get something from the dimension formula and the examples. It has enough original construction and application to merit sending to a serious referee rather than a desk reject. I would recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The paper defines a cochain complex C^•_mix(M) whose cohomology controls deformations of the mixed associator of a module category M over a k-linear monoidal category C. It shows that C^•_mix(M) is isomorphic to the Davydov-Yetter complex of the representation functor ρ: C → End(M). Using prior results on DY cohomology, it proves that if C and M are finite then H^•_mix(M) ≅ Ext^•_{Z(C),C}(1, A_M) for the adjunction between the Drinfeld center Z(C) and C, where A_M is the adjoint algebra of M. This yields a dimension formula for H^n_mix(M) in terms of Hom spaces in Z(C) and proves H^{>0}_mix(C) = 0. The paper further shows A_M is the full center of an algebra realizing M, establishes a generalized Ocneanu rigidity for monoidal functors with coefficients (applicable to non-exact, non-finite C-module categories over fusion categories C with dim(C) ≠ 0), and works out explicit examples for comodule algebras over finite-dimensional Hopf algebras, including new continuous families over Sweedler's Hopf algebra.

Significance. If the central identifications are verified, the results link mixed-associator deformations directly to relative Ext groups in the Drinfeld center, providing computable dimension formulas and rigidity statements that extend classical results like Ocneanu rigidity. The explicit Hopf algebra examples and the treatment of non-exact module categories add concrete value to the literature on tensor categories and their deformations.

major comments (2)

The section establishing the isomorphism C^•_mix(M) ≅ DY complex of ρ: the paper asserts this identification (invoked to apply the main theorem of arXiv:2411.19111) but the provided text does not exhibit an explicit chain map or direct comparison of differentials arising from the pentagonator/module-associator data versus the monoidal structure on ρ; any mismatch in signs or higher coherence would invalidate the subsequent Ext isomorphism and all dimension/vanishing formulas derived from it.
The paragraph deriving the dimension formula for H^n_mix(M) from the Ext groups: this step inherits the identification risk above and is load-bearing for the claim that the cohomology is 'isomorphic to the relative Ext groups'; an independent verification or reference to the precise chain-level isomorphism in the prior work is needed.

minor comments (2)

Notation: the adjoint algebra A_M is introduced without an early explicit definition or diagram; adding a short definition in the introduction would improve readability.
The statement of generalized Ocneanu rigidity could include a precise formulation of the 'with coefficients' version and a pointer to where the proof differs from the classical case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed major comments. We agree that the isomorphism requires a more explicit treatment to be fully rigorous and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: The section establishing the isomorphism C^•_mix(M) ≅ DY complex of ρ: the paper asserts this identification (invoked to apply the main theorem of arXiv:2411.19111) but the provided text does not exhibit an explicit chain map or direct comparison of differentials arising from the pentagonator/module-associator data versus the monoidal structure on ρ; any mismatch in signs or higher coherence would invalidate the subsequent Ext isomorphism and all dimension/vanishing formulas derived from it.

Authors: We agree that an explicit chain map and direct comparison of differentials (including signs and coherence) should be exhibited rather than asserted. In the revised manuscript we will add a dedicated subsection that constructs the chain map explicitly from the definitions of the mixed cochain complex (using the module associator data) to the Davydov-Yetter complex of the representation functor ρ, verifies that it is a chain map by direct computation on generators, and checks compatibility with the higher coherence conditions. This will make the identification fully rigorous and allow safe invocation of the main theorem from arXiv:2411.19111. revision: yes
Referee: The paragraph deriving the dimension formula for H^n_mix(M) from the Ext groups: this step inherits the identification risk above and is load-bearing for the claim that the cohomology is 'isomorphic to the relative Ext groups'; an independent verification or reference to the precise chain-level isomorphism in the prior work is needed.

Authors: We will revise the paragraph to include an explicit reference to the precise chain-level isomorphism theorem in arXiv:2411.19111 and add a short independent verification sketch showing how the dimension formula for the relative Ext groups in the Drinfeld center translates directly to the stated formula for dim H^n_mix(M) in terms of Hom spaces in Z(C). This will make the derivation self-contained once the chain map is established. revision: yes

Circularity Check

1 steps flagged

Central Ext isomorphism obtained via self-cited DY cohomology results from arXiv:2411.19111

specific steps

self citation load bearing [Abstract]
"We show that C^•_mix(M) is isomorphic to the Davydov-Yetter (DY) complex of the representation functor ρ : C → End(M). Using our previous results on DY cohomology (arXiv:2411.19111), we prove that if C and M are finite then the cohomology H^•_mix(M) is isomorphic to the relative Ext groups Ext^•_{Z(C),C}(1, A_M)"

The isomorphism H^•_mix(M) ≅ Ext^•_{Z(C),C}(1, A_M) is obtained directly by substituting the main theorem of the authors' own prior paper arXiv:2411.19111 for the cohomology of the DY complex; the present manuscript provides no independent chain-map verification or re-proof of that substitution, making the central claim rest on the overlapping-author citation.

full rationale

The paper independently defines the mixed cochain complex C^•_mix(M) and claims an isomorphism to the Davydov-Yetter complex of the representation functor ρ. The load-bearing step converting this to the relative Ext isomorphism with the adjoint algebra A_M invokes the authors' prior results on DY cohomology without re-derivation here. This matches the self-citation load-bearing pattern but leaves independent content in the mixed-complex setup and the claimed isomorphism to DY. No self-definitional reductions, fitted predictions, or ansatz smuggling appear in the provided derivation chain. The finite assumptions and dimension formulas inherit the self-citation but do not collapse the entire argument by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on standard axioms of monoidal categories and module categories (associativity up to coherent isomorphism, finite-dimensionality assumptions) plus the prior DY cohomology results; no new free parameters are introduced, but the adjoint algebra A_M and the mixed complex itself are defined within the paper.

axioms (2)

standard math Standard coherence axioms for monoidal categories and module categories (associators, unitors satisfying pentagon and triangle identities).
Invoked throughout the setup of the mixed associator and the representation functor.
domain assumption Finite-dimensionality of C and M to obtain the Ext isomorphism and dimension formula.
Explicitly stated as the condition under which the main identification holds.

invented entities (2)

Mixed cochain complex C^•_mix(M) no independent evidence
purpose: Controls deformations of the mixed associator
Newly defined complex whose cohomology is the main object of study.
Adjoint algebra A_M of the module category M no independent evidence
purpose: Object in the Drinfeld center used to express the cohomology via relative Ext
Defined as the full center of an algebra realizing M; central to the isomorphism.

pith-pipeline@v0.9.0 · 5712 in / 1761 out tokens · 68630 ms · 2026-05-13T22:10:58.520847+00:00 · methodology

discussion (0)

Reference graph

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