arxiv: 2604.20527 · v1 · submitted 2026-04-22 · 🧮 math.RT

Recognition: unknown

Representation Cohomology of a Small Category

Daniel Solch, Henri Riihimaki, Markus Klemetti, Ran Levi

Pith reviewed 2026-05-09 22:52 UTC · model grok-4.3

classification 🧮 math.RT

keywords representation cohomologysmall categoriessimplicial objects in Catnerve of a categoryGrothendieck groupsmodule categoriescochain complexes

0 comments

The pith

Any small category yields a representation cohomology by turning the simplices of its nerve into a simplicial object in Cat and taking cohomology of the resulting Grothendieck group complex.

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

The paper associates to every small category C a simplicial object in the category of small categories whose n-th level has objects given by the n-simplices of the nerve of C. Representation cohomology is defined as the cohomology of the cochain complex formed from the Grothendieck groups of isomorphism classes of modules over the categories appearing at each level. The authors then examine the basic properties of this cohomology, including its behavior on certain subobjects, and carry out explicit computations in favourable cases.

Core claim

To any small category C we associate a simplicial object C_• in Cat where for each n the objects of the level-n category are the simplices of the nerve of C. For a field k, the Grothendieck groups of isomorphism classes of kC_n-modules form a cochain complex whose cohomology is the representation cohomology of the simplicial object and of selected subobjects.

What carries the argument

The simplicial object in Cat built from the nerve of C, whose levels supply the categories whose module Grothendieck groups assemble into the cochain complex defining representation cohomology.

If this is right

The representation cohomology is defined for any small category and for any field k.
Subobjects of the associated simplicial object in Cat produce their own cochain complexes and therefore their own representation cohomologies.
Basic properties of the cohomology follow directly from the simplicial structure and the functoriality of Grothendieck groups.
Explicit computations of the cohomology are possible when the category is simple enough that the nerve and its module categories are tractable.

Where Pith is reading between the lines

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

The construction supplies a uniform way to attach cohomological invariants to objects studied in representation theory of categories.
One could compare the resulting groups with classical invariants such as group cohomology when C is a group or with simplicial cohomology when C arises from a space.
The same method might be applied to other coefficient systems beyond Grothendieck groups of modules, such as K-theory spectra or derived categories.

Load-bearing premise

The face and degeneracy maps between the module categories induce maps on Grothendieck groups that compose to differentials satisfying d squared equals zero.

What would settle it

A concrete small category C for which two consecutive induced maps on Grothendieck groups compose to a nonzero homomorphism.

read the original abstract

Let $C_\bullet$ be a simplicial object in the category $Cat$ of small categories. For a field $k$, taking the Grothendieck groups of isomorphism classes of $kC_n$-modules gives rise to a cochain complex, whose cohomology, which we refer to as representation cohomology, is the object studied in this article. In particular, to any small category $C$, we associate a simplicial object in $Cat$, where for each $n\ge 0$ the objects of the level $n$ category are the simplices of the nerve of $C$. The basic properties of the resulting representation cohomology of these simplicial objects and certain subobjects are then studied in detail. We present some general theoretical computations in favourable cases.

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 / 3 minor

Summary. The paper defines representation cohomology for a simplicial object C_• in Cat: for a field k, the terms of the cochain complex are the Grothendieck groups of isomorphism classes of kC_n-modules, with differentials induced by the face maps of C_•. It specializes to the simplicial object associated to any small category C, where the objects of the level-n category are the n-simplices of the nerve of C, and studies the basic properties of the resulting cohomology groups together with computations in favorable cases.

Significance. If the construction is well-defined, the work supplies a new cohomological invariant for small categories that combines their simplicial nerve structure with representation theory over a field. The use of Grothendieck groups ensures the construction is invariant under isomorphism of representations, and the simplicial identities guarantee d² = 0 once the induced maps on Grothendieck groups are shown to be well-defined. This may link to existing invariants in algebraic K-theory or homological algebra of categories.

minor comments (3)

§2 (or the section defining the simplicial object): the precise description of the morphisms in the level-n category whose objects are the n-simplices of the nerve should be stated explicitly, as the current wording leaves open whether the morphisms are induced by the face and degeneracy maps of the nerve or defined separately.
The notation for the Grothendieck group (e.g., K_0(kC_n) or G_0(kC_n)) is used inconsistently in the early sections; a single consistent symbol should be fixed and used throughout.
In the computations of favorable cases, the paper should include at least one fully worked small example (e.g., C the category with two objects and one non-identity morphism) showing the explicit cochain complex and its cohomology groups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for the positive assessment of its significance as a new cohomological invariant combining simplicial structure with representation theory. We note the recommendation for minor revision and will incorporate any necessary adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; construction is definitional and self-contained

full rationale

The paper defines a simplicial object C_• in Cat from a small category C by taking the n-simplices of the nerve of C as the objects of the level-n category. Representation cohomology is then defined directly as the cohomology of the cochain complex whose terms are the Grothendieck groups of isomorphism classes of kC_n-modules. The differentials are induced by the face maps of C_•, which are functors and therefore induce exact functors on the abelian categories of k-linear representations, yielding well-defined maps on Grothendieck groups; the simplicial identities ensure d² = 0 by construction. No equations reduce the output cohomology to a fitted parameter or to a quantity defined in terms of itself. No self-citations are invoked as load-bearing for the central claim or uniqueness, and the subsequent study of basic properties follows as consequences of the definition without circular reduction. The derivation chain is independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard fact that Grothendieck groups of modules over a category algebra can be assembled into a cochain complex when the underlying data is simplicial. No free parameters, ad-hoc axioms, or new postulated entities beyond the definition of the cohomology itself are visible in the abstract.

axioms (1)

domain assumption Grothendieck groups of isomorphism classes of kC_n-modules form a cochain complex for the chosen simplicial object in Cat
The abstract asserts that taking these groups gives rise to a cochain complex whose cohomology is the object of study.

invented entities (1)

representation cohomology no independent evidence
purpose: Cohomology of the cochain complex built from Grothendieck groups of kC_n-modules
This is the central new invariant defined and studied in the paper.

pith-pipeline@v0.9.0 · 5427 in / 1509 out tokens · 59557 ms · 2026-05-09T22:52:21.454135+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Bauer, M

U. Bauer, M. Botnan, S. Oppermann, and J. SteenCotorsion torsion triples and the representation theory of filtered hierarchical clustering, Advances in Mathematics, 369, pp. 1–51, 2020

2020
[2]

Foundations of Differential Calculus for modules over posets

J. Brodzki, R. Levi, H. Riihimäki,Foundations of differential calculus for modules over posets, arXiv:2307.02444, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[3]

Klemetti, D

M. Klemetti, D. Sölch, Representation Cohomology Calculator
[4]

Caputi, H

L. Caputi, H. Riihimäki,On reachability categories, persistence, and commuting algebras of quivers, Theory and Applications of Categories, V ol. 41, (2024), No. 12, pp 426–448

2024
[5]

A DrozdTame and wild matrix problems, Representations and quadratic forms (Institute of Mathe- matics, Academy of Sciences, Ukrainian SSR, Kiev, 1979); Amer

Yu. A DrozdTame and wild matrix problems, Representations and quadratic forms (Institute of Mathe- matics, Academy of Sciences, Ukrainian SSR, Kiev, 1979); Amer. Math. Soc. Transl, 128, pp. 39–74, 1986

1979
[6]

Derksen, J

H. Derksen, J. Weyman, An Introduction to Quiver Representations, Graduate Studies in Mathematics 184, AMS, 2017

2017
[7]

E. G. Escolar and Y . Hiraoka,Persistence modules on commutative ladders of finite type, Discrete and Computational Geometry, 55 pp. 100–157, 2016

2016
[8]

Hepworth, E

R. Hepworth, E. Roff,The Reachability Homology of a Directed Graph, International Mathematics Research No- tices, 2025 (3) 1–18

2025
[9]

Goerss, J

P. Goerss, J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics, 174, Birkhäuser, 1999

1999
[10]

Klemetti, R

M. Klemetti, R. Levi, H. RiihimäkiThe gradient for modules over posets generated by trees, To Appear. 40
[11]

MacLane.Categories for the working Mathematician 2nd Ed., Springer, 1998

S. MacLane.Categories for the working Mathematician 2nd Ed., Springer, 1998

1998
[12]

Mitchell,Rings with several objects,Advances in Mathematics, 8, pp

B. Mitchell,Rings with several objects,Advances in Mathematics, 8, pp. 1–161, 1972

1972
[13]

Quillen, AlgebraicK-theory I, Lecture notes in mathematics 341 (1973), 77–139

D. Quillen, AlgebraicK-theory I, Lecture notes in mathematics 341 (1973), 77–139

1973
[14]

Schiffler.Quiver Representations

R. Schiffler.Quiver Representations. CMS Books in Mathematics. Canadian Mathematical Society, 2014

2014
[15]

Webb.An introduction to the representations and cohomology of categories, Group Representation Theory (EPFL Press), pp

P. Webb.An introduction to the representations and cohomology of categories, Group Representation Theory (EPFL Press), pp. 149–173, 2007

2007
[16]

C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, 38, CUP, (1994)

1994
[17]

Xu,Representations of categories and their applications, Journal of Algebra, 317 pp

F. Xu,Representations of categories and their applications, Journal of Algebra, 317 pp. 153–183, 2007. Institute ofMathematics, University ofAberdeen, Aberdeen, UK Email address:markus.o.klemetti@gmail.com Institute ofMathematics, University ofAberdeen, Aberdeen, UK Email address:r.levi@abdn.ac.uk Nordita, StockholmUniversity, Stockholm, Sweden Email addr...

2007