Projective and anomalous representations of categories and their linearizations

Chetan Vuppulury; Domenico Fiorenza

REVIEW 3 major objections 3 minor 12 references

Projective and anomalous representations of categories and their linearizations

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes a three-way equivalence making every anomaly-linearization of a category, including the fermionic field-theory construction, one universal categorical operation.

desk verdict Real new categorical triangle, but the Eilenberg–Watts leg is built on an ill-typed cocontinuity assumption and needs a finite-dimensional variant before the abstract can be taken at face value. read the letter →

arxiv 2506.01521 v2 pith:FWMIITOJ submitted 2025-06-02 math.CT math-phmath.MPmath.QAmath.RT

Domenico Fiorenza , Chetan Vuppulury This is my paper

classification math.CTmath-phmath.MPmath.QAmath.RT

keywords anomalousrepresentationsprojectivelaxhomotopypullback2-vectorspacescocontinuousfunctorscentralextensionsanomalylinearization∞-categories

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's goal is to show that the standard fix for anomalous functorial field theories—extend the bordism category so that the anomaly becomes a linear action—is not ad hoc but forced by a universal categorical construction. Given any category C and any anomaly functor J from C into the 2-category of algebras, bimodules, and intertwiners, the paper builds an extension C^J of C and a smaller subcategory C^J_ST inside it. It then proves a three-way equivalence: anomalous representations of C with anomaly J are the same as Vect-linear, colimit-preserving functors out of C^J, and these are the same as ordinary linear functors out of C^J_ST on which J acts by scalars. This generalizes the classical correspondence between projective representations of a group and linear representations of its central extension, and it specializes to the bordism construction used to linearize anomalous conformal field theories.

What carries the argument

The load-bearing object is the extension C^J, defined as the lax homotopy pullback of J against a point: an object is a pair (X,L_X) with L_X a right module over J(X), and a morphism is a morphism of C decorated by a compatible module homomorphism. Inside it sits the distinguished subcategory C^J_ST, the full subcategory on the objects (X,J(X)); since an algebra is a module over itself, a morphism there is a morphism of C equipped with a pointing, i.e., a chosen element of the bimodule J(f). The argument is carried by the recognition theorem for additive colimit-preserving functors between module categories—such a functor is necessarily tensoring with a module—applied fiberwise over each object of C. That theorem is what lets the paper pass from an anomalous representation Z to the functor L_X ↦ L_X ⊗_{J(X)} Z(X), and back, and it is also what isolates the 'J acts as scalars' condition as exactly the image of Vect_K-linear functors under restriction to C^J_ST.

What would settle it

Choose a finite group G and a nontrivial 2-cocycle $\alpha$, and compute explicitly the two sides of the triangle for C=BG: the category of projective representations of G of class $\alpha$ and the category of J_alpha-acts-as-scalars functors on (BG)^{J_alpha}_{ST}. A direct count of isomorphism classes, for example with the quaternion group and its nontrivial 2-cocycle, would either confirm the bijection or expose the missing cocontinuity hypothesis; any mismatch would falsify Proposition 8.8 as stated.

Watch

Extended reading notes

Core claim

The central discovery is a commuting triangle of equivalences (Proposition 8.8) for any functor J:C→2Vect: the category of anomalous representations of C with anomaly J, the category of Vect_K-linear additive colimit-preserving functors out of C^J, and the category of linear functors on C^J_ST on which J acts as scalars are all equivalent. The extension C^J is the lax homotopy fiber of J, and C^J_ST is its full subcategory on the objects (X,J(X)), where J(X) is regarded as a right module over itself. Restriction from C^J to C^J_ST is an equivalence onto the 'J acts as scalars' functors, and the inverse is built by tensoring fibers with the representing module supplied by the additive-cocontinuous recognition theorem. In the single-object case C=BG with J=J_alpha, this triangle reduces to the classical equivalence between projective representations of G of class $\alpha$ and linear representations of the central extension on which K^* acts by scalars; in the bordism case it reproduces the fermionic-anomaly linearization of conformal field theories.

Load-bearing premise

The equivalence rests on assuming the functors out of C^J preserve all colimits on each fiber category of finite-dimensional modules, a condition the paper does not prove is well defined there; if that fails, the triangle as stated is too strong.

Editorial extensions

If this is right

For every anomaly J, anomaly cancellation can be performed uniformly: replace C by C^J, and anomalous representations become ordinary Vect_K-linear functors.
The 'J acts as scalars' condition is a complete invariant: a linear functor on C^J_ST extends to C^J if and only if it satisfies it.
The classical projective-representation theorem is a special case: with C=BG and J the anomaly associated to a 2-cocycle alpha, the triangle is the correspondence with linear representations of the K^*-central extension on which K^* acts by scalars.
The fermionic and degree-n anomaly extensions of conformal bordism categories are recovered as special cases of the same triangle.
The same construction extends to super vector spaces and Hilbert-space targets, so the linearization mechanism is not specific to finite-dimensional vector spaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical test the paper leaves implicit: to check whether a proposed linearization of an anomalous theory is complete, verify the 'J acts as scalars' condition on the ST subcategory rather than constructing the full extension by hand.
If the cocontinuity hypothesis proves too strong for finite-dimensional module categories, the equivalence likely survives in modified form for functors preserving only the colimits those categories actually have; the classical central-extension theorem for infinite-dimensional representations is a natural place to test this.
The same lax-pullback pattern suggests a hierarchy of higher anomalies: replacing Vect and 2Vect by n-vector spaces should linearize anomalous n-representations in the same way, with the extension playing the role of a higher central extension.
The right-extension reading of the twisted group algebra in the appendix hints that the whole construction can be rephrased as: the anomaly J determines a universal target algebra, and anomalous representations are modules over it; making that precise for general C would give an even more compact formulation of the triangle.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: this paper has a genuine new claim—a triangle of equivalences linking anomalous representations of a category C with anomaly J, Vect_K-linear functors on the extension C^J, and representations of the Stolz–Teichner subcategory C^J_ST where J acts by scalars. The group case recovers the classical projective-representation/central-extension theorem, and the proof is unusually explicit, with diagrams unwound and the special case checked. That is real work, and it deserves credit.

What is actually new is the construction itself: the extension C^J and the subcategory C^J_ST, and the proof that the three categories are equivalent. I did not find this triangle in the cited literature. The authors also honestly flag that the full infinity-categorical version is deferred to forthcoming work.

The soft spot is exactly where the stress-test note lands. Definition 6.9 defines Hom_{Vect_K}(C^J, Vect_K) as additive and cocontinuous functors 'over C', but the fibers Mod_{J(X)} are finite-dimensional module categories over a finite-dimensional algebra. They are not cocomplete—no infinite direct sums—so 'preserving small colimits' is not well-typed for these fibers. Then Prop 8.6 applies the classical Eilenberg–Watts theorem to these fibers to identify E|_X(L) with L ⊗_{J(X)} E|_X(J(X)). The cited theorem (Eil60, Wat60) is for categories of all modules, which are cocomplete. What the proof needs is a finite-dimensional variant: any additive K-linear functor preserving finite colimits (or cokernels) from Mod_A^fd to Vect_K is naturally isomorphic to -⊗_A M. No such lemma is stated or proved. So the proof of the right-hand equivalence in the triangle has a genuine gap, and the abstract's unqualified 'Vect-linear functors' overstates what is established; the additive/cocontinuous hypothesis is omitted.

That said, the flaw is likely repairable. The other two identities in the triangle—Prop 8.5 and Prop 8.7—do not use Eilenberg–Watts, and the group example checks out. If the authors add the finite-right-exact hypothesis to Def 6.9 and prove the finite-dimensional EW lemma, the triangle should stand.

Who this is for: people working on anomalies in functorial field theory and higher representation theory. They will find the categorical scaffolding useful, but they should not take the abstract at face value until the EW leg is fixed. It deserves a serious referee, with a request to repair Def 6.9, prove the missing finite-dimensional Eilenberg–Watts statement, and soften the abstract.

Referee Report

3 major / 3 minor

Summary. This paper develops a categorical framework for projective and anomalous representations. For a functor J:C→2Vect_K, it defines an anomalous representation as a lax homotopy commutative triangle relative to the terminal map, constructs the lax homotopy fiber C^J, and a Stolz–Teichner subcategory C^J_ST. The main result, Proposition 8.8, asserts a commuting triangle of equivalences among (a) anomalous representations with anomaly J, (b) Vect_K-linear additive cocontinuous functors C^J→Vect_K, and (c) linear representations of C^J_ST on which J acts as scalars. The group case with J_α recovers the classical equivalence between projective representations of class α and linear representations of the central extension G^α on which K^* acts as scalars, and the appendix recovers the twisted group algebra as a Kan extension.

Significance. If valid, Proposition 8.8 gives a universal linearization of anomalies, subsuming the Stolz–Teichner Clifford-linear construction and the classical theory of projective group representations. The paper's strengths are its explicit and detailed proofs, its careful unwindings of simplicial definitions, the verification of the group special case, and the absence of hidden fitting parameters. The main theorem is proved from categorical universal properties and checked against an external standard (Eilenberg–Watts); however, the key Eilenberg–Watts input is not established in the finite-dimensional setting used here, so the central claim is presently conditional on a missing lemma.

major comments (3)

[§6, Definition 6.9; §8, Lemma 8.3 and Proposition 8.6] The hypothesis 'additive and cocontinuous (i.e., preserving small colimits) over C' in Definition 6.9 is not well-formed for the fiber categories used in the proof. By §2, Mod_{J(X)} is the category of finite-dimensional right J(X)-modules; for J(X)=K this is Vect_K^fd, which has no infinite direct sums or filtered colimits. Hence the restrictions E|_X of Remark 6.5 have no small colimits to preserve, and the classical Eilenberg–Watts theorem cited in §8.1 does not apply as stated. This matters because Lemma 8.3 and both parts of Proposition 8.6 use Eilenberg–Watts to identify E|_X with –⊗_{J(X)} E|_X(J(X)); Proposition 8.6 is exactly the right-hand equivalence in Proposition 8.8. Please replace 'small colimits' by 'finite colimits' (or otherwise make the finite-dimensional setting precise) and state and prove the needed finite-dimensional Eilenberg–Watts statement, including the version for natural transformations used in the second half of the proof of Proposition 8.6.
[Abstract and §1] The abstract's claim (i) and the introduction's corresponding sentence present anomalous representations as equivalent to 'Vect-linear functors E:C^J→Vect' without the additive and cocontinuous restriction. The body's Definition 6.9 defines Hom_{Vect_K}(C^J,Vect_K) only for functors that are additive and cocontinuous over C, and Proposition 6.10 lands in that subcategory. The unqualified headline is therefore stronger than what is proved and should be amended (e.g., 'additive and cocontinuous Vect-linear functors'), with the same correction made wherever the result is summarized.
[Appendix, Lemma 8.10] Lemma 8.10 justifies the right Kan extension (8.6) by asserting that 'BG is small and 2Vect_K is complete'. Completeness of the Morita 2-category of finite-dimensional algebras, bimodules, and intertwiners is neither proved nor referenced, so the lemma as stated rests on an unverified input. For the group case the extension is constructed explicitly in Lemma 8.12 and Corollary 8.14 does not need the general completeness claim; please either give a proof or precise citation for completeness, or restrict Lemma 8.10 to the explicit construction for BG.

minor comments (3)

[Throughout] There are numerous typographical slips, e.g., 'invesigate' in the abstract, 'abritray 8-category', 'subteltiles', 'construtions', and 'well kown'; also the symbol '8' appears in place of '∞' in several places (e.g., §1 and Remark 5.10). These should be corrected.
[§8, after Lemma 8.3] The notation Hom_K(C^J_ST,Vect_K) is used before being explicitly defined; please insert a formal definition (the full subcategory of Hom(C^J_ST,Vect_K) on functors for which each λ_X is a K-algebra map).
[§8, Proposition 8.6] The proof writes the composite 'E_{Z_{E_ST}}' without first defining it; please introduce notation for the functor obtained by applying Proposition 6.10 to Z_{E_{ST}} and then restricting, so that the displayed comparisons are unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central triangle is proved from the external Eilenberg–Watts theorem and recovers the classical group case as a special case, not as an input.

full rationale

The derivation chain in Propositions 6.10, 8.5, 8.6, and 8.8 is self-contained relative to standard external results. The main equivalence Hom_{Vect_K}(C^J, Vect_K) ≅ Hom_K(C^J_ST, Vect_K) ≅ anomalous representations is established by explicit constructions (E_Z, Z_F, restriction to C^J_ST) and by two applications of the classical Eilenberg–Watts theorem [Eil60, Wat60], not by assuming the conclusion. The classical group statement about projective representations and central extensions is recovered in Example 8.9 and the Appendix as a specialization, which is a check of the framework rather than a load-bearing input. The only self-referential items are the acknowledgement that the article is based on the thesis [Vup25] and Remark 5.10 deferring a fully general (∞,n)-categorical version to [Vup26]; neither is used to justify Proposition 8.8 or the anomaly-linearization claim. Any concern about the applicability of Eilenberg–Watts to finite-dimensional module categories is a mathematical-correctness question about hypotheses, not a circularity of the argument.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted and no new physical entities are postulated. The central claim depends on standard higher-categorical machinery and on the Eilenberg-Watts theorem in the finite-dimensional module setting; both are treated as black boxes. The appendix's algebra K^J[C] is a further consequence whose existence for arbitrary 2-categories is delegated to the literature.

assumptions (3)

standard math Lax homotopy pullbacks and infinity-categorical universal properties behave as in Lurie's Higher Topos Theory and Kerodon.
Used throughout to define C^J, C^J//BK^*, and the functor E_Z in Sections 6 through 8.
domain assumption Eilenberg-Watts theorem applies to additive cocontinuous functors Mod_A -> Vect_K, including for the finite-dimensional module categories used here.
Load-bearing in Propositions 8.5 and 8.6; the finite-dimensional setting is not discussed, so this is an assumption about the theorem's scope.
domain assumption For small C and J: C -> 2Vect_K, the right Kan extension of J along C -> * exists, yielding an algebra K^J[C] unique up to Morita equivalence.
Used in the appendix to recover twisted group algebras; for arbitrary 2-categories it relies on Johnson-Freyd and Reutter and is not proved in this paper.

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Cite this review

Pith. "Pith review of Projective and anomalous representations of categories and their linearizations." pith.science (2026). https://pith.science/paper/FWMIITOJ

@misc{pith2026250601521,
  author       = {Pith},
  title        = {Pith review of: Projective and anomalous representations of categories and their linearizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWMIITOJ}},
  note         = {Machine review of arXiv:2506.01521}
}

abstract

We invesigate the relation between projective and anomalous representations of categories, and show how to any anomaly $J\colon \mathcal{C}\to 2\mathrm{Vect}$ one can associate an extension $\mathcal{C}^J$ of $\mathcal{C}$ and a subcategory $\mathcal{C}^J_{\mathrm{ST}}$ of $\mathcal{C}^J$ with the property that: (i) anomalous representations of $\mathcal{C}$ with anomaly $J$ are equivalent to $\mathrm{Vect}$-linear functors $E\colon \mathcal{C}^J\to \mathrm{Vect}$, and (ii) these are in turn equivalent to linear representations of $\mathcal{C}^J_{\mathrm{ST}}$ where "$J$ acts as scalars". This construction, inspired by and generalizing the technique used to linearize anomalous functorial field theories in the physics literature, can be seen as a multi-object version of the classical relation between projective representations of a group $G$, with given $2$-cocycle $\alpha$, and linear representations of the central extension $G^\alpha$ of $G$ associated with $\alpha$.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Abstract description of some basic functors

Samuel Eilenberg. Abstract description of some basic functors. J. Indian Math. Soc. (N.S.) , 24:231--234 (1961), 1960

work page 1961

[2] [2]

Minimal nondegenerate extensions

Theo Johnson-Freyd and David Reutter. Minimal nondegenerate extensions. J. Amer. Math. Soc. , 37(1):81--150, 2024. https://doi.org/10.1090/jams/1023 doi:10.1090/jams/1023

work page doi:10.1090/jams/1023 2024

[3] [3]

The chiral anomaly of the free fermion in functorial field theory

Matthias Ludewig and Saskia Roos. The chiral anomaly of the free fermion in functorial field theory. Ann. Henri Poincar\' e , 21(4):1191--1233, 2020. https://doi.org/10.1007/s00023-020-00893-6 doi:10.1007/s00023-020-00893-6

work page doi:10.1007/s00023-020-00893-6 2020

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Phd thesis (in preparation)

Raphael Schmidpeter. Phd thesis (in preparation). 2025

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Michael A. Shulman. Constructing symmetric monoidal bicategories. 2020. URL: https://arxiv.org/abs/1004.0993

arXiv 2020