Some Remarks About Integral Categories

Max Zinchenko; Yaroslav Kopylov

arxiv: 2606.21248 · v1 · pith:RXR4P7SAnew · submitted 2026-06-19 · 🧮 math.CT

Some Remarks About Integral Categories

Yaroslav Kopylov , Max Zinchenko This is my paper

Pith reviewed 2026-06-26 12:50 UTC · model grok-4.3

classification 🧮 math.CT

keywords integral categoriesone-sided integralityright integralleft integralcategory theorycategorical algebraasymmetric properties

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The pith

An example separates right integrality from left integrality in categories.

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

Integral categories appear in categorical foundations of algebra and functional analysis. The paper supplies new criteria for deciding one-sided and two-sided integrality. It then presents the first known concrete category that satisfies the right-integral condition while failing the left-integral condition. This shows the two sides are logically independent. A reader sees the work as establishing that integrality need not be symmetric.

Core claim

For the first time in the literature, an example is given of a right but not left integral category, together with new criteria for one- and two-sided integrality.

What carries the argument

The distinction between left and right integrality, witnessed by an explicit category that meets the right condition but not the left.

Load-bearing premise

The standard definitions and familiar criteria for integrality from the existing literature are taken as given and applied correctly to the new example.

What would settle it

A direct check showing that the constructed category either fails right integrality or satisfies left integrality would refute the central claim.

read the original abstract

We consider integral categories, a class of categories that gained importance in the last three decades in the categorical foundations of algebra and functional analysis. We discuss some familiar criteria for integrality and prove new criteria for one- and two-sided integrality. For the first time in the literature, an example is given of a right but not left integral category.

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's real contribution is the first explicit example of a right-but-not-left integral category, plus a few new criteria; the rest is mostly discussion of known material.

read the letter

The headline result is the construction of a category that is right integral but fails to be left integral. The abstract states this example is new to the literature, and the authors also supply some additional criteria for one-sided and two-sided integrality.

That distinction is the part worth noting. In the subfield of categorical foundations for algebra, separating left and right versions can affect how certain verification or construction arguments are set up, so a concrete counterexample to the implication is useful to have on record.

The paper does the straightforward job of recalling familiar criteria, stating the new ones, and exhibiting the example. No hidden parameters or circular reductions appear in the claims.

The limitation is scope. Everything stays inside category theory proper, and the example's construction details are not visible from the abstract alone. If the example turns out to be artificial or hard to generalize, its practical value drops. The proofs of the new criteria are asserted but not sketched here, so their length and tightness cannot be judged yet.

This is for specialists already working on integral categories or one-sided properties in categorical algebra. A reader outside that narrow circle will not find much to take away.

It is worth sending to referees. The central claim is a first-time example rather than a routine extension, and the topic has a small but established literature, so a careful check of the construction and proofs makes sense.

Referee Report

0 major / 2 minor

Summary. The manuscript examines integral categories in the categorical foundations of algebra and functional analysis. It reviews familiar criteria for integrality, proves new criteria for one- and two-sided integrality, and supplies the first explicit example in the literature of a right integral but not left integral category.

Significance. If the example and criteria are correct, the work is significant for demonstrating that left and right integrality are distinct, thereby clarifying an asymmetry not previously exhibited by concrete construction. The new criteria add verifiable tools for checking integrality. The paper supplies an explicit example construction resting on standard definitions, which is a strength as it is in principle reproducible and falsifiable.

minor comments (2)

The abstract states that proofs and an example exist but supplies no section references or brief indications of the constructions used; adding such pointers would improve navigability without altering the technical content.
The title is generic; a subtitle or more precise phrasing could better signal the focus on new criteria and the asymmetric example.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper takes standard definitions and criteria for integral categories as given from the existing literature, then proves new criteria and constructs an explicit example of a right-but-not-left integral category. No equations reduce a claimed prediction to a fitted input by construction, no self-citation chain is load-bearing for the central result, and no ansatz or uniqueness theorem is smuggled in via prior author work. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned or required by the abstract; the work relies on existing definitions of integral categories from the prior three decades of literature.

pith-pipeline@v0.9.1-grok · 5563 in / 993 out tokens · 19348 ms · 2026-06-26T12:50:19.087905+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references

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C. B˘ anic˘ a and N. Popescu,Sur les cat´ egories pr´ eab´ eliennes, Rev. Roumaine Math. Pures Appl.10(1965), 621–633

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Br¨ ustle, S

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2021
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Eckmann and P

B. Eckmann and P. J. Hilton,Exact couples in an abelian category, J. Algebra3(1966), 38–87

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S. Hassoun, A. Shah and S.-A. Wegner (2021)Examples and non-examples of integral cat- egories and the admissible intersection property, Cahiers Topologie G´ eom. Diff´ erentielle Cat´ eg. 62, no. 3, 329-354

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Kopylov,Exact couples in a Ra˘ ıkov semi-abelian category, Cah

Ya. Kopylov,Exact couples in a Ra˘ ıkov semi-abelian category, Cah. Topol. G´ eom. Diff´ er. Cat´ eg.45(2004), no. 3, 162–178

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Kopylov and S.-A

Ya. Kopylov and S.-A. Wegner,On the notion of a semi-abelian category in the sense of Palamodov, Appl. Categor. Struct.20(2012), no. 5, 531–541

2012
[9]

Kopylov and S.-A

Ya. Kopylov and S.-A. Wegner,Exact couples in semiabelian categories revisited, J. Algebra 414(2014), 264–270. doi10.1016/j.jalgebra.2014.04.030

2014
[10]

Lawson and S

M. Lawson and S. A. Wegner,The category of Silva spaces is not integral, Homology Ho- motopy Appl.25, 367–374 (2023)

2023
[11]

W. S. Massey,Exact couples in algebraic topology. I, II, Ann. of Math. (2)56(1952), 363–396

1952
[12]

Nakaoka,General heart construction for twin torsion pairs on triangulated categories, J

H. Nakaoka,General heart construction for twin torsion pairs on triangulated categories, J. Algebra374(2013), 195–215

2013
[13]

Richman and E

F. Richman and E. A. Walker,Ext in pre-Abelian categories, Pacific J. Math.71(1977), no. 2, 521–535

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[14]

Rump,Almost abelian categories, Cahiers Topologie G´ eom

W. Rump,Almost abelian categories, Cahiers Topologie G´ eom. Diff´ erentielle Cat´ eg.42 (2001), no. 3, 163–225

2001
[15]

RumpA counterexample to Raikov’s conjecture, Bull

W. RumpA counterexample to Raikov’s conjecture, Bull. Lond. Math. Soc.40(2008), no. 6, 985–994

2008
[16]

Schneiders,Quasi-Abelian categories and sheaves, M´ em

J.-P. Schneiders,Quasi-Abelian categories and sheaves, M´ em. Soc. Math. Fr., Nouv. S´ er. 76, 1999

1999
[17]

Sieg and S.-A

D. Sieg and S.-A. Wegner,Maximal exact structures on additive categories, Math. Nachr. 284(2011), no. 16, 2093–2100

2011
[18]

A. V. Yakovlev,Homological algebra in pre-abelian categories, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)94(1979), 131–141, English translation: J. Soviet Math., 19(1982), 1060–1067. Yaroslav Kopylov Sobolev Institute of Mathematics 4 Koptyug A ve. 630090, Novosibirsk, Russia Email address:yakop@math.nsc.ru Max Zinchenko Novosibirsk Sta...

1979

[1] [1]

B˘ anic˘ a and N

C. B˘ anic˘ a and N. Popescu,Sur les cat´ egories pr´ eab´ eliennes, Rev. Roumaine Math. Pures Appl.10(1965), 621–633

1965

[2] [2]

Br¨ ustle, S

T. Br¨ ustle, S. Hassoun, and A. Tattar,Intersections, sums, and the Jordan-H¨older property for exact categories, J. Pure Appl. Algebra225(2021), no. 11, 106724

2021

[3] [3]

A. B. Buan, B. R. Marsh,From triangulated categories to module categories via localisation. II: Calculus of fractions, J. Lond. Math. Soc., II. Ser.86, 152–170 (2012); corrigendum87, 643 (2013)

2012

[4] [4]

Eckmann and P

B. Eckmann and P. J. Hilton,Exact couples in an abelian category, J. Algebra3(1966), 38–87

1966

[5] [5]

Hassoun, A

S. Hassoun, A. Shah and S.-A. Wegner (2021)Examples and non-examples of integral cat- egories and the admissible intersection property, Cahiers Topologie G´ eom. Diff´ erentielle Cat´ eg. 62, no. 3, 329-354

2021

[6] [6]

G. M. Kelly,Monomorphisms, epimorphisms, and pull-backs, J. Austral. Math. Soc.9 (1969), 124–142

1969

[7] [7]

Kopylov,Exact couples in a Ra˘ ıkov semi-abelian category, Cah

Ya. Kopylov,Exact couples in a Ra˘ ıkov semi-abelian category, Cah. Topol. G´ eom. Diff´ er. Cat´ eg.45(2004), no. 3, 162–178

2004

[8] [8]

Kopylov and S.-A

Ya. Kopylov and S.-A. Wegner,On the notion of a semi-abelian category in the sense of Palamodov, Appl. Categor. Struct.20(2012), no. 5, 531–541

2012

[9] [9]

Kopylov and S.-A

Ya. Kopylov and S.-A. Wegner,Exact couples in semiabelian categories revisited, J. Algebra 414(2014), 264–270. doi10.1016/j.jalgebra.2014.04.030

2014

[10] [10]

Lawson and S

M. Lawson and S. A. Wegner,The category of Silva spaces is not integral, Homology Ho- motopy Appl.25, 367–374 (2023)

2023

[11] [11]

W. S. Massey,Exact couples in algebraic topology. I, II, Ann. of Math. (2)56(1952), 363–396

1952

[12] [12]

Nakaoka,General heart construction for twin torsion pairs on triangulated categories, J

H. Nakaoka,General heart construction for twin torsion pairs on triangulated categories, J. Algebra374(2013), 195–215

2013

[13] [13]

Richman and E

F. Richman and E. A. Walker,Ext in pre-Abelian categories, Pacific J. Math.71(1977), no. 2, 521–535

1977

[14] [14]

Rump,Almost abelian categories, Cahiers Topologie G´ eom

W. Rump,Almost abelian categories, Cahiers Topologie G´ eom. Diff´ erentielle Cat´ eg.42 (2001), no. 3, 163–225

2001

[15] [15]

RumpA counterexample to Raikov’s conjecture, Bull

W. RumpA counterexample to Raikov’s conjecture, Bull. Lond. Math. Soc.40(2008), no. 6, 985–994

2008

[16] [16]

Schneiders,Quasi-Abelian categories and sheaves, M´ em

J.-P. Schneiders,Quasi-Abelian categories and sheaves, M´ em. Soc. Math. Fr., Nouv. S´ er. 76, 1999

1999

[17] [17]

Sieg and S.-A

D. Sieg and S.-A. Wegner,Maximal exact structures on additive categories, Math. Nachr. 284(2011), no. 16, 2093–2100

2011

[18] [18]

A. V. Yakovlev,Homological algebra in pre-abelian categories, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)94(1979), 131–141, English translation: J. Soviet Math., 19(1982), 1060–1067. Yaroslav Kopylov Sobolev Institute of Mathematics 4 Koptyug A ve. 630090, Novosibirsk, Russia Email address:yakop@math.nsc.ru Max Zinchenko Novosibirsk Sta...

1979