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arxiv: 2507.06124 · v3 · pith:GZLXUTPEnew · submitted 2025-07-08 · 🧮 math.CT · math.LO· math.RA

Coherent and ideal actions in ideally exact categories

Manuel Mancini , Giuseppe Metere , Federica Piazza This is my paper

Pith reviewed 2026-05-19 05:46 UTC · model grok-4.3

classification 🧮 math.CT math.LOmath.RA
keywords ideally exact categoriesinternal coherent actioninternal ideal actionsemidirect productunital actionscategory theorycoherent actions
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The pith

Every ideal action is coherent in ideally exact categories, with the converse true in key contexts.

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

The paper defines internal coherent actions and internal ideal actions in ideally exact categories. These notions extend different features of unital actions that rings and algebras perform. The main result establishes that every ideal action is coherent. The converse holds in some relevant ideally exact contexts. The work also connects the notions to Janelidze's semidirect product in these categories.

Core claim

In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze's notion of semidirect product in ideally exact categories is analysed.

What carries the argument

Internal coherent actions and internal ideal actions, which generalize aspects of unital actions from rings and algebras within ideally exact categories.

If this is right

  • Every ideal action satisfies the defining properties of a coherent action.
  • The two notions of action coincide in certain ideally exact categories.
  • Semidirect products in ideally exact categories can be related directly to ideal and coherent actions.
  • Actions in abstract categorical settings can be studied by transferring results between the two notions.

Where Pith is reading between the lines

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

  • These definitions may apply to other categories that share similar exactness properties with ideally exact ones.
  • Concrete examples in categories of modules or groups could be worked out to test the generality.
  • Further links might exist to other categorical constructions involving actions or products.

Load-bearing premise

The definitions of internal coherent action and internal ideal action correctly generalize the intended aspects of unital actions of rings and algebras within ideally exact categories.

What would settle it

An ideally exact category containing an ideal action that is not coherent, or a relevant context where the converse fails.

read the original abstract

In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze's notion of semidirect product in ideally exact categories is analysed.

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 / 2 minor

Summary. In the context of ideally exact categories, the paper introduces the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. It proves that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze's notion of semidirect product in ideally exact categories is analysed.

Significance. If the results hold, this contributes to categorical algebra by extending the study of actions to ideally exact categories in a way that captures key features from classical ring and algebra theory. A notable strength is the explicit verification that the new definitions, introduced via universal properties, reduce to the classical unital actions on the category of rings and on module categories; the ideal-implies-coherent implication is derived directly from the axioms of ideally exact categories without additional hidden assumptions.

minor comments (2)
  1. The abstract states that the converse holds in 'some relevant ideally exact contexts' but does not name the extra structure (e.g., pointedness or a class of regular epimorphisms); adding one sentence would improve orientation for readers.
  2. In the sections introducing the internal actions, the universal properties are stated explicitly, but a short remark confirming that the reduction to the classical ring case follows immediately from the definitions would help readers verify the generalisation without reconstructing the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. The report correctly identifies the introduction of internal coherent and ideal actions in ideally exact categories, the proof that every ideal action is coherent (with the converse holding in relevant contexts), and the analysis of connections to Janelidze's semidirect products. We appreciate the emphasis on the explicit verification that these notions recover classical unital actions in the categories of rings and modules.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines internal coherent and ideal actions via explicit universal properties in ideally exact categories, proves the ideal-implies-coherent implication directly from the category axioms, and establishes the converse only under explicitly stated extra assumptions. These steps do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The connection to Janelidze semidirect products references prior external work without circular dependency on the present results. The derivation chain is self-contained against the stated axioms and classical special cases.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the background theory of ideally exact categories and on Janelidze's prior definition of semidirect products. No free parameters or new postulated entities are introduced; the contribution consists of new definitions and proofs inside an established framework.

axioms (1)
  • domain assumption Ideally exact categories possess the exactness properties required for the internal notions of action to be well-defined.
    Invoked implicitly when the new actions are defined inside ideally exact categories.

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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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    Relation between the paper passage and the cited Recognition theorem.

    We introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts.

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matches
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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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.

Reference graph

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