Hochschild cohomology for functors on linear symmetric monoidal categories
Pith reviewed 2026-05-24 06:03 UTC · model grok-4.3
The pith
Hochschild cohomology extends to the category of linear functors on symmetric monoidal categories via natural hom constructions.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In the symmetric monoidal category F of R-linear functors on an essentially small symmetric monoidal R-enriched category, the natural hom constructions allow one to define Hochschild cohomology for modules over monoids in F, and these definitions yield the usual long exact sequences and other classical properties.
What carries the argument
Natural hom-objects in the functor category F that behave like the classical Hom functor and enable the formation of cochain complexes for cohomology.
If this is right
- The standard properties of Hochschild cohomology, including long exact sequences, hold for modules over monoids in F.
- Cohomology groups can be defined and computed directly from the natural hom constructions without additional structure.
- The theory applies to any essentially small symmetric monoidal category enriched in R-Mod.
- Classical Hochschild cohomology appears as the special case when the base category is a single ring.
Where Pith is reading between the lines
- The same natural-hom approach may extend to other cohomology theories such as cyclic or André-Quillen cohomology inside F.
- Applications to invariants of representations or tensor categories become available once the base category is taken to be a representation category.
- The construction supplies a template for lifting any cohomology theory that relies on a symmetric monoidal structure and a hom functor to the functor setting.
Load-bearing premise
The functor category F must itself be symmetric monoidal and its natural hom-objects must satisfy the usual adjunction and exactness properties of classical Hom.
What would settle it
A concrete monoid in F together with a module whose associated cochain complex fails to produce the expected long exact sequence in cohomology when computed via the natural homs.
read the original abstract
Let $R$ be a commutative ring with unit. We develop a Hochschild cohomology theory in the category $\mathcal{F}$ of linear functors defined from an essentially small symmetric monoidal category enriched in $R$-Mod, to $R$-Mod. The category $\mathcal{F}$ is known to be symmetric monoidal too, so one can consider monoids in $\mathcal{F}$ and modules over these monoids, which allows for the possibility of a Hochschild cohomology theory. The emphasis of the article is in considering natural hom constructions appearing in this context. These homs, together with the abelian structure of $\mathcal{F}$ lead to nice definitions and provide effective tools to prove the main properties and results of the classical Hochschild cohomology theory.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a Hochschild cohomology theory in the category F of linear functors from an essentially small symmetric monoidal R-enriched category to R-Mod. It uses the fact that F is itself symmetric monoidal (via Day convolution) to form monoids and modules inside F, and relies on natural hom constructions together with the abelian structure of F to define the cohomology and recover the standard properties of classical Hochschild cohomology.
Significance. If the natural hom-objects in F can be shown to inherit the required adjointness, exactness, and balancing properties from the classical Hom functor, the construction would furnish a direct categorical extension of Hochschild cohomology to functor categories. This could supply new computational tools in representation theory and enriched category theory, particularly when the domain category carries additional structure.
major comments (2)
- [Abstract] Abstract: the central claim that 'natural hom constructions ... lead to nice definitions and provide effective tools to prove the main properties' is asserted without any explicit cocycle definitions, bar-resolution construction, or verification that the resulting groups satisfy the expected universal properties of Hochschild cohomology (e.g., derivation from Ext in the monoid-module category).
- [Abstract] The load-bearing step—that the natural hom-objects in F behave sufficiently like the classical Hom functor to carry the usual cohomology constructions—receives no explicit check of adjointness or exactness properties inside F; without this verification the transfer of classical arguments remains formal.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying points where the presentation of the constructions can be made more explicit. We address the two major comments below and will revise the manuscript accordingly to strengthen the exposition.
read point-by-point responses
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Referee: [Abstract] Abstract: the central claim that 'natural hom constructions ... lead to nice definitions and provide effective tools to prove the main properties' is asserted without any explicit cocycle definitions, bar-resolution construction, or verification that the resulting groups satisfy the expected universal properties of Hochschild cohomology (e.g., derivation from Ext in the monoid-module category).
Authors: We agree that the abstract is concise and does not spell out these elements. In the body of the paper the Hochschild cochain complex is defined via the internal hom (natural hom) in the closed monoidal category F, and the bar resolution is constructed in the category of modules over a monoid in F. To address the concern directly we will expand the introduction with an explicit cocycle-level definition, include the bar-resolution construction in a dedicated subsection, and add a proposition verifying that the resulting cohomology is isomorphic to the appropriate Ext groups in the monoid-module category, thereby confirming the expected universal property. revision: yes
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Referee: [Abstract] The load-bearing step—that the natural hom-objects in F behave sufficiently like the classical Hom functor to carry the usual cohomology constructions—receives no explicit check of adjointness or exactness properties inside F; without this verification the transfer of classical arguments remains formal.
Authors: The category F is symmetric monoidal closed under Day convolution, so the natural hom is the internal hom and satisfies the required adjunction by the definition of the closed structure. Exactness follows from the fact that F is abelian and the internal hom preserves the relevant limits and colimits in each variable. Nevertheless, we acknowledge that an explicit verification of these properties in the specific context of the Hochschild construction would make the transfer of classical arguments fully transparent. We will therefore insert a short lemma that records the adjointness and exactness properties of the natural hom in F and shows how they are used in the cohomology definitions. revision: yes
Circularity Check
No significant circularity; direct transfer of classical constructions
full rationale
The paper develops Hochschild cohomology in the functor category F by invoking its known symmetric monoidal structure (via Day convolution) and natural hom-objects together with the abelian structure of F to carry over the classical definitions and proofs. No equations, fitted parameters, or self-citations are shown to reduce the claimed cohomology groups or their properties to quantities already present by construction inside the paper; the argument is presented as a straightforward adaptation of standard bar-resolution and Ext techniques to this enriched setting. The load-bearing assumption that the hom-objects inherit the necessary adjointness and exactness properties is taken as part of the setup rather than derived from the target result.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The category F of R-linear functors from an essentially small symmetric monoidal R-linear category to R-Mod is itself symmetric monoidal.
Reference graph
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discussion (0)
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