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arxiv: 2505.14545 · v2 · submitted 2025-05-20 · 🧮 math.AG · math.CT· math.QA

Tensor-Hochschild complex

Slava Pimenov , Angel Toledo This is my paper

Pith reviewed 2026-05-22 13:53 UTC · model grok-4.3

classification 🧮 math.AG math.CTmath.QA
keywords monoidal dg-categoryTensor-Hochschild complexDavydov-Yetter complexGerstenhaber-Schack complexoperadic E2-cohomologyquasi-bialgebradeformation theorybialgebra representations
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The pith

A Tensor-Hochschild complex controls joint deformations of a monoidal dg-category and its tensor product.

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

The authors start with an arbitrary monoidal dg-category and build a complex whose cohomology is meant to classify deformations that change both the underlying category and the monoidal operation at the same time. When the category is semisimple the new complex reduces to the known Davydov-Yetter complex. In the special case of modules over a commutative algebra the cohomology computes the operadic E2-cohomology of the algebra, while for representations of an associative bialgebra the construction recovers the Gerstenhaber-Schack complex and extends it to quasi-bialgebras.

Core claim

Starting from any monoidal dg-category (C, ⊗), the Tensor-Hochschild complex is constructed so that its cohomology classifies simultaneous deformations of the dg-category and of the monoidal structure. The complex reduces to the Davydov-Yetter complex when C is semisimple, computes the operadic E2-cohomology of A when C is the category of A-modules for a commutative algebra A, and recovers the Gerstenhaber-Schack complex when C is the representation category of an associative bialgebra, while also providing a direct generalization to quasi-bialgebras.

What carries the argument

The Tensor-Hochschild complex, built directly from the data of an arbitrary monoidal dg-category to govern joint deformations of the category and its tensor product.

If this is right

  • For semisimple monoidal dg-categories the new complex coincides with the Davydov-Yetter complex.
  • When the monoidal category is the category of modules over a commutative algebra, the complex computes the operadic E2-cohomology of that algebra.
  • When the monoidal category is the representation category of an associative bialgebra, the complex recovers the Gerstenhaber-Schack complex.
  • The same construction supplies a generalization of the Gerstenhaber-Schack complex to the setting of quasi-bialgebras.

Where Pith is reading between the lines

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

  • The construction may supply a uniform way to treat deformations that mix categorical and operadic data across different examples.
  • It could be tested on concrete quasi-bialgebras arising from quantum groups to see whether new deformation classes appear.
  • The method might extend to higher monoidal structures or to settings in derived algebraic geometry where both the category and its tensor product are deformed simultaneously.

Load-bearing premise

That the data of an arbitrary monoidal dg-category is enough to define a single complex whose cohomology classifies all joint deformations of the category and its monoidal structure.

What would settle it

A direct computation, for a concrete semisimple monoidal dg-category, showing that the cohomology of the Tensor-Hochschild complex differs from the cohomology of the Davydov-Yetter complex.

read the original abstract

Let $(\mathcal{C}, \otimes)$ be a monoidal dg-category. We construct a complex controlling the deformation of the monoidal structure on $\mathcal{C}$ together with the deformation of the underlying dg-category itself. We show that in the case of a semisimple category $\mathcal{C}$ it reduces to the Davydov-Yetter complex. Furthermore, we study this complex in several special cases, in particular, in the case of the category of $A$-modules over a commutative algebra $A$ we obtain a complex computing operadic $E_2$-cohomology of $A$. And in the case of the category of representations of an associative bialgebra we recover the Gerstenhaber-Schack complex. In the latter case our construction can be considered as a generalization of the Gerstenhaber-Schack complex to quasi-bialgebras.

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

2 major / 2 minor

Summary. The paper constructs the Tensor-Hochschild complex for an arbitrary monoidal dg-category (C, ⊗). The central claim is that the cohomology of this complex classifies simultaneous deformations of the monoidal structure and the underlying dg-category. The authors verify the construction by explicit reductions to the Davydov-Yetter complex when C is semisimple, to the operadic E₂-cohomology complex for the category of modules over a commutative algebra A, and to the Gerstenhaber-Schack complex for representations of an associative bialgebra, with an extension of the latter to quasi-bialgebras.

Significance. If the construction and reductions hold, the Tensor-Hochschild complex would supply a unified cohomological framework for joint deformations in monoidal dg-categories, generalizing several classical complexes and extending them to quasi-bialgebras. The explicit consistency checks against known special cases strengthen the overall claim and could facilitate applications in categorical deformation theory and related areas of algebraic geometry.

major comments (2)
  1. [Main construction (around the definition of the complex)] The manuscript asserts that the Tensor-Hochschild complex controls joint deformations for arbitrary monoidal dg-categories, yet the supporting argument appears to rest primarily on the special-case reductions rather than a direct general proof that the cohomology classifies the deformations; a theorem stating the general classification result with a sketch of the obstruction theory would make the central claim load-bearing.
  2. [Section on bialgebra representations] In the reduction to the Gerstenhaber-Schack complex for bialgebra representations, the explicit chain map or quasi-isomorphism establishing the equivalence is not detailed enough to verify that both the monoidal and underlying-category deformation parameters are preserved; this is load-bearing for the generalization claim to quasi-bialgebras.
minor comments (2)
  1. [Introduction] The introduction would benefit from a short table comparing the Tensor-Hochschild complex with the Davydov-Yetter, Gerstenhaber-Schack, and E₂ complexes in terms of generators and differentials.
  2. Notation for the differential in the general complex could be made more uniform to avoid confusion with standard Hochschild differentials.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestions. We address the major comments point by point below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Main construction (around the definition of the complex)] The manuscript asserts that the Tensor-Hochschild complex controls joint deformations for arbitrary monoidal dg-categories, yet the supporting argument appears to rest primarily on the special-case reductions rather than a direct general proof that the cohomology classifies the deformations; a theorem stating the general classification result with a sketch of the obstruction theory would make the central claim load-bearing.

    Authors: We agree that a direct general statement would make the central claim more self-contained. The Tensor-Hochschild complex is constructed precisely so that its cochains parametrize infinitesimal deformations of both the monoidal structure and the underlying dg-category, with the differential encoding the necessary compatibility conditions. In the revised manuscript we will add an explicit theorem stating the general classification result, together with a brief sketch of the obstruction theory obtained by analyzing the differential on the complex. revision: yes

  2. Referee: [Section on bialgebra representations] In the reduction to the Gerstenhaber-Schack complex for bialgebra representations, the explicit chain map or quasi-isomorphism establishing the equivalence is not detailed enough to verify that both the monoidal and underlying-category deformation parameters are preserved; this is load-bearing for the generalization claim to quasi-bialgebras.

    Authors: We accept that additional detail is needed for verification. In the revised version we will expand the section on bialgebra representations to include an explicit description of the chain map (or quasi-isomorphism) and verify directly that it identifies the deformation parameters for both the monoidal structure and the underlying category. This will also clarify the extension to quasi-bialgebras. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper defines the Tensor-Hochschild complex directly from an arbitrary monoidal dg-category (C, ⊗) without presupposing the special-case complexes. It then verifies by explicit reduction that the cohomology matches the Davydov-Yetter complex for semisimple C, the operadic E₂-cohomology for A-modules, and the Gerstenhaber-Schack complex for bialgebra representations. These reductions function as consistency checks on the independently constructed complex rather than as definitional inputs or self-citations that bear the central claim. No load-bearing step reduces by construction to fitted parameters, prior self-citations, or ansatzes imported from the authors' own work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions of monoidal dg-categories and semisimple categories; no free parameters or new postulated entities are introduced beyond the complex itself.

axioms (1)
  • domain assumption C is a monoidal dg-category equipped with a tensor product ⊗.
    This is the explicit starting data for the construction given in the first sentence of the abstract.

pith-pipeline@v0.9.0 · 5668 in / 1230 out tokens · 46753 ms · 2026-05-22T13:53:01.551962+00:00 · methodology

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