On the equivalence between the existence of $n$-kernels and $n$-cokernels

Vitor Gulisz; Wolfgang Rump

arxiv: 2510.23369 · v2 · submitted 2025-10-27 · 🧮 math.CT · math.RA· math.RT

On the equivalence between the existence of n-kernels and n-cokernels

Vitor Gulisz , Wolfgang Rump This is my paper

Pith reviewed 2026-05-18 03:50 UTC · model grok-4.3

classification 🧮 math.CT math.RAmath.RT

keywords n-kernelsn-cokernelspreadditive categoriesidempotent complete categoriesweak kernelsweak cokernelsglobal dimensionmodule categories

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The pith

In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do.

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

The paper establishes an equivalence: under the stated conditions on a category, the ability to form sequences of n kernels is equivalent to the ability to form sequences of n cokernels. This holds for every nonnegative integer n and is proved directly without heavy machinery. A reader would care because the result equates left-sided and right-sided homological constructions in abstract settings. It immediately supplies simpler proofs that global dimensions agree for certain pairs of module categories.

Core claim

If an idempotent complete preadditive category possesses weak kernels and weak cokernels, then for any nonnegative integer n the category admits n-kernels if and only if it admits n-cokernels. The proof proceeds by constructing explicit correspondences between kernel sequences and cokernel sequences that preserve the necessary exactness properties.

What carries the argument

The equivalence between the existence of n-kernels and n-cokernels, realized by relating successive kernel constructions to successive cokernel constructions inside the preadditive structure.

If this is right

Global dimension of the category of right modules equals global dimension of the category of left modules in the two cases treated as corollaries.
The n-kernel and n-cokernel properties are interchangeable without additional assumptions beyond those given.
Homological invariants that can be defined via either kernels or cokernels become symmetric by this equivalence.

Where Pith is reading between the lines

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

The result may let researchers transfer finiteness or exactness conditions from one side of a duality to the other in non-abelian settings.
Specific examples such as the category of modules over a ring could be checked directly to confirm the n=2 case matches known facts about projective and injective dimensions.
The elementary character of the proof suggests similar direct arguments could relate other paired notions such as n-projectives and n-injectives.

Load-bearing premise

The category under study is idempotent complete and preadditive and already has weak kernels together with weak cokernels.

What would settle it

Exhibit one idempotent complete preadditive category that has weak kernels and weak cokernels yet possesses n-kernels for some n while lacking n-cokernels for the same n.

read the original abstract

We give an elementary proof of the statement that if an idempotent complete preadditive category has weak kernels and weak cokernels, then it has $n$-kernels if and only if it has $n$-cokernels, where $n$ is a nonnegative integer. As a consequence, elementary proofs of two results concerning the equality between the global dimensions of certain right and left module categories are obtained.

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 gives a clean inductive proof that n-kernels exist iff n-cokernels do in idempotent-complete preadditive categories with weak (co)kernels, plus simpler arguments for some global-dimension equalities.

read the letter

The main takeaway is that this equivalence holds by a direct induction that treats kernels and cokernels symmetrically. The base cases fall out from the weak (co)kernel hypothesis, and each inductive step builds the next diagram from the previous one while using idempotent completeness to split the idempotents that arise when gluing sequences. Both directions are proved the same way, without leaning on duality or extra additivity assumptions. That is the actual new piece: an elementary argument that shortens some existing proofs about global dimensions in right and left module categories. The paper does this part well. The construction is explicit, the induction is straightforward, and the consequences are stated clearly as corollaries rather than as the main claim. No load-bearing gaps appear in the outline. The result is narrow by design. It applies only inside idempotent-complete preadditive categories that already have weak kernels and weak cokernels, so it does not claim to work in more general or less structured settings. That limitation is stated up front and does not undermine the stated theorem. The citation pattern is light and appropriate for a short proof note. This is for people working on homological algebra in preadditive or exact categories who want shorter arguments for dimension comparisons. A reading group focused on module categories or categorical homological algebra would get value from it. The paper is self-contained enough to deserve referee time.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that if an idempotent-complete preadditive category possesses weak kernels and weak cokernels, then it has n-kernels if and only if it has n-cokernels, for any nonnegative integer n. The argument proceeds by induction on n, constructing successive (co)kernel diagrams from the weak (co)kernel hypothesis and using idempotent completeness to split the idempotent endomorphisms that arise. As a consequence, elementary proofs are given for the equality of global dimensions between certain right and left module categories.

Significance. If the result holds, it supplies a useful duality-type statement for homological properties in preadditive categories that need not be additive or abelian, allowing symmetric treatment of kernels and cokernels. The elementary inductive proof, which treats both directions symmetrically and reduces the base cases directly to the weak (co)kernel assumption, is a clear strength. The applications to global-dimension equalities in module categories add concrete value and demonstrate the result's utility.

minor comments (2)

[§3] §3 (inductive step): the construction of the (n+1)-kernel diagram from the n-kernel and the weak kernel is described clearly, but a short diagram or explicit sequence of morphisms for the n=2 case would make the gluing step easier to follow for readers new to the notation.
[§4] §4 (applications): the two results on global dimensions are stated as corollaries, but the precise module categories (e.g., right vs. left modules over a ring with certain properties) are referenced rather than recalled; a one-sentence reminder of the relevant setup would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and encouraging report, including the recognition of the result's significance as a duality-type statement and the value of the elementary inductive proof. We appreciate the recommendation for minor revision and will address any editorial or presentational suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper gives an elementary inductive proof establishing equivalence between n-kernels and n-cokernels in an idempotent-complete preadditive category already equipped with weak kernels and weak cokernels. The construction proceeds symmetrically in both directions by successively building (co)kernel diagrams from the weak hypotheses and splitting resulting idempotents via the idempotent-completeness assumption; base cases (n=0,1) reduce directly to the given weak (co)kernels without any fitted parameters, self-definitional loops, or load-bearing self-citations. The derivation is therefore self-contained against the stated hypotheses and does not reduce any central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the domain assumptions of idempotent completeness, preadditivity, and the existence of weak kernels and weak cokernels; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The category is idempotent complete and preadditive and possesses weak kernels and weak cokernels.
This is the explicit hypothesis of the main equivalence statement.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. Let C be an idempotent complete preadditive category that has weak kernels and weak cokernels... Then C has n-kernels if and only if C has n-cokernels.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.