Riguet and Generalized Congruences on a Category: Relationships and Applications

Enric Cosme Ll\'opez; Juan Climent Vidal; Ra\'ul Ruiz Mora

arxiv: 2505.15767 · v5 · pith:L5TPM5S7new · submitted 2025-05-21 · 🧮 math.CT

Riguet and Generalized Congruences on a Category: Relationships and Applications

Juan Climent Vidal , Enric Cosme Ll\'opez , Ra\'ul Ruiz Mora This is my paper

Pith reviewed 2026-05-22 13:45 UTC · model grok-4.3

classification 🧮 math.CT

keywords Riguet congruencesgeneralized congruencesalgebraic latticedirected-complete posetScott continuous morphismrelative adjunctionclassified categoriescategory theory

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The pith

Riguet congruences on any category form a bounded directed-complete poset while generalized congruences form an algebraic lattice linked by a Scott-continuous morphism.

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

The paper proves that for any category C the collection of Riguet congruences ordered by inclusion is a bounded directed-complete poset and the collection of generalized congruences is an algebraic lattice. It constructs a Scott-continuous morphism connecting these two ordered structures. The same morphism lifts to relative adjunctions between the categories RCgr(C) and GCgr(C) and between the categories RCCat and GCCat of Riguet-classified and generalized-classified categories. The work further characterizes functors that are full and surjective on objects via regular epimorphisms and strong generalized congruences, and relates the constructions to Manes' framework and Grothendieck fibrations. A reader would care because these lattice and adjunction results organize quotients, epimorphisms, and factorization data uniformly across arbitrary categories.

Core claim

For a category C, the set RCgr(C) of all Riguet congruences ordered by inclusion is a bounded directed-complete ordered set, while the set GCgr(C) of all generalized congruences is an algebraic lattice. A Scott-continuous morphism exists between these ordered sets. The lattice-theoretic facts lift to relative adjunctions between the categories RCgr(C) and GCgr(C) and between RCCat and GCCat. Full and surjective-on-objects functors are characterized in terms of regular epimorphisms, extremal epimorphisms, and strong and regular generalized congruences.

What carries the argument

The Scott-continuous morphism from the bounded directed-complete poset RCgr(C) to the algebraic lattice GCgr(C) that induces relative adjunctions between the associated categories.

If this is right

Full functors that are surjective on objects admit characterizations via regular epimorphisms, extremal epimorphisms, and strong or regular generalized congruences.
The lattice and poset structures lift to relative adjunctions between RCgr(C) and GCgr(C) as well as between RCCat and GCCat.
The wide subcategory of full morphisms in RCCat relates to GCCat inside Manes' framework of K-objects with structure and to Grothendieck fibrations.
Riguet congruences apply to constructions in several mathematical fields outside pure category theory.

Where Pith is reading between the lines

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

The Scott-continuous bridge may allow transfer of completeness or algebraic properties between Riguet and generalized congruences when concrete categories are substituted for the arbitrary C.
The relative adjunctions suggest a possible comparison with factorization systems or reflective subcategories built from congruence lattices.
Applications listed in the paper could be tested by computing the two congruence posets explicitly in familiar categories such as sets or groups.

Load-bearing premise

Riguet congruences and generalized congruences are well-defined and closed under the operations needed to form the stated ordered structures in every category without size or completeness assumptions.

What would settle it

A concrete category C in which the set of Riguet congruences ordered by inclusion fails to be directed-complete or in which the proposed morphism from RCgr(C) to GCgr(C) is not Scott-continuous.

read the original abstract

We investigate Riguet congruences and generalized congruences on a category, focusing on their interrelations from both lattice-theoretic and category-theoretic perspectives. We also characterize functors that are full and surjective on objects in terms of regular epimorphisms, extremal epimorphisms and in terms of strong and regular generalized congruences. On the lattice-theoretic side, we prove that for a category $\mathsf{C}$, the set $\mathrm{RCgr}(\mathsf{C})$ of all Riguet congruences, ordered by inclusion, is a bounded directed-complete ordered set, while the set $\mathrm{GCgr}(\mathsf{C})$ of all generalized congruences is an algebraic lattice. We establish a bridge between these structures via a Scott continuous morphism. From a category-theoretic standpoint, we lift these results to relative adjunctions between the categories $\mathsf{RCgr}(\mathsf{C})$ and $\mathsf{GCgr}(\mathsf{C})$ associated to the above ordered sets, as well as between the categories $\mathsf{RCCat}$, of Riguet classified categories, and $\mathsf{GCCat}$, of generalized classified categories. Furthermore, within Manes' framework of categories of $\mathsf{K}$-objects with structure, we investigate the relationship between the wide subcategory $\mathsf{RCCat}_{\mathrm{full}}$ of $\mathsf{RCCat}$, whose morphisms are the full morphisms of $\mathsf{RCCat}$, and $\mathsf{GCCat}$, relating these constructions to the Grothendieck theory of fibrations. Finally, we present applications of Riguet congruences across various mathematical fields.

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 organizes Riguet and generalized congruences into dcpos and lattices with a Scott bridge and relative adjunctions, but the claims for arbitrary categories hit set-size problems.

read the letter

The key takeaway is that this paper establishes lattice-theoretic properties for Riguet congruences and generalized congruences on any category, proves a Scott-continuous connection between them, and lifts the whole setup to relative adjunctions between the corresponding categories. It also gives characterizations of full surjective functors in terms of epimorphisms and these congruences, and explores links to Manes' K-object categories and Grothendieck fibrations. What stands out as new is the combination of the directed-complete poset for RCgr and the algebraic lattice for GCgr, together with the Scott bridge and the relative adjunctions. The applications to classified categories and the wide subcategory of full morphisms add some category-theoretic flavor. The functor characterizations are standard in style but applied here to these specific congruence notions. The paper does well in bringing lattice and category perspectives together on these two kinds of congruences. The claims in the abstract hang together without obvious internal contradictions. The soft spot is the handling of arbitrary categories. The results treat RCgr(C) and GCgr(C) as sets that form a dcpo and an algebraic lattice, but for a large category these collections are proper classes. Without a smallness assumption or universe framework, the statements about bounded directed-complete ordered sets and algebraic lattices do not go through in standard set theory. This is the main issue flagged in the stress test, and it affects the central claims. This paper targets researchers in categorical algebra who work with quotients, congruences, and fibrations. Someone already familiar with Riguet congruences or Manes' framework would get the most out of the connections and applications. It is solid enough in its core ideas to deserve a serious referee, provided the size questions get sorted out. I recommend putting it through peer review with attention to the foundational assumptions on the category C.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates Riguet congruences and generalized congruences on an arbitrary category C. It proves that the poset RCgr(C) of Riguet congruences is bounded and directed-complete, that GCgr(C) of generalized congruences forms an algebraic lattice, and that a Scott-continuous map exists between them. These structures are lifted to relative adjunctions between the categories RCgr(C) and GCgr(C) as well as between RCCat and GCCat. The paper also characterizes full and object-surjective functors via regular/extremal epimorphisms and strong/regular generalized congruences, relates the constructions to Grothendieck fibrations within the framework of K-objects, and lists applications across mathematical fields.

Significance. If the lattice and adjunction results hold under appropriate size hypotheses, the work supplies a concrete bridge between two congruence notions in category theory, together with functor characterizations that are standard in form but here tied explicitly to the new lattice structures. The explicit Scott-continuous morphism and the lift to relative adjunctions between the associated categories constitute the main technical contribution and could support further work on classified categories and fibrations.

major comments (2)

[Abstract and §1] Abstract and §1 (Introduction): the statements that RCgr(C) is a bounded directed-complete poset and GCgr(C) an algebraic lattice are asserted for an arbitrary category C with no smallness or universe restriction stated. For a large C these collections are proper classes, so the dcpo and algebraic-lattice claims are not literally true in ZFC; the subsequent Scott-continuous map and relative adjunctions inherit the same foundational gap. This is load-bearing for every lattice-theoretic and adjunction claim in the paper.
[§3] §3 (Lattice-theoretic results): the proof that GCgr(C) is an algebraic lattice relies on the existence of arbitrary joins and meets within the collection; without an explicit hypothesis that C is small (or that one works inside a Grothendieck universe), the joins may not exist as sets, rendering the algebraic-lattice property formally invalid.

minor comments (2)

[Throughout] Notation: the manuscript alternates between boldface, sans-serif and plain fonts for category names (C, RCgr(C), etc.); a uniform convention would improve readability.
[Applications section] The applications section lists several fields but does not indicate which concrete theorems are being applied; a short pointer from each application back to the relevant lattice or adjunction result would strengthen the exposition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the foundational size issue in our lattice-theoretic claims. We agree that explicit smallness hypotheses are required for the statements about RCgr(C) and GCgr(C) to hold rigorously in ZFC. In the revised manuscript we will add the assumption that C is small (or that we work inside a fixed Grothendieck universe) at the appropriate places, including the abstract, §1 and §3. This will ensure all subsequent results on Scott-continuous maps and relative adjunctions rest on solid set-theoretic ground without changing the core mathematical content.

read point-by-point responses

Referee: [Abstract and §1] The statements that RCgr(C) is a bounded directed-complete poset and GCgr(C) an algebraic lattice are asserted for an arbitrary category C with no smallness or universe restriction stated. For a large C these collections are proper classes, so the dcpo and algebraic-lattice claims are not literally true in ZFC; the subsequent Scott-continuous map and relative adjunctions inherit the same foundational gap.

Authors: We fully accept this observation. The original text does not restrict C to be small, which is necessary for RCgr(C) and GCgr(C) to be sets. In the revision we will insert the explicit hypothesis that C is small (or that all constructions take place inside a Grothendieck universe) in the abstract and in §1. Under this hypothesis the collections become sets, the directed-completeness and algebraic-lattice properties hold, and the Scott-continuous bridge together with the relative adjunctions between RCgr(C) and GCgr(C) (and between RCCat and GCCat) are well-defined. We will also add a brief remark on the size assumption in the introduction. revision: yes
Referee: [§3] The proof that GCgr(C) is an algebraic lattice relies on the existence of arbitrary joins and meets within the collection; without an explicit hypothesis that C is small (or that one works inside a Grothendieck universe), the joins may not exist as sets, rendering the algebraic-lattice property formally invalid.

Authors: We agree that the proof in §3 presupposes the relevant collections are sets. We will add the smallness hypothesis for C at the beginning of §3 and ensure it is stated globally. With this addition, arbitrary joins and meets of generalized congruences exist as sets, the algebraic-lattice axioms are satisfied, and the remainder of the proof proceeds unchanged. The same hypothesis will be referenced when the Scott-continuous map and the relative adjunctions are constructed later in the paper. revision: yes

Circularity Check

0 steps flagged

No circularity; claims derived directly from definitions

full rationale

The paper's central claims—that RCgr(C) is a bounded directed-complete poset and GCgr(C) an algebraic lattice for arbitrary C, with a Scott-continuous bridge inducing relative adjunctions—are presented as theorems proved from the standard definitions of Riguet congruences and generalized congruences. No step reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation whose content is itself unverified. The derivation relies on explicit closure properties under joins, meets, and the Scott topology, remaining self-contained within the given category-theoretic framework without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper works entirely within standard category theory; it introduces no new free parameters, no ad-hoc axioms beyond the usual axioms of categories and lattices, and no invented entities such as new particles or dimensions.

axioms (2)

standard math Standard axioms of category theory (composition, identities, associativity) and the definitions of Riguet congruence and generalized congruence are taken as given.
Invoked throughout the statements about RCgr(C) and GCgr(C) being ordered sets with the listed completeness properties.
domain assumption The poset of congruences under inclusion admits the required suprema and infima needed for directed-completeness and algebraicity.
Required for the lattice-theoretic claims in the abstract.

pith-pipeline@v0.9.0 · 5834 in / 1644 out tokens · 36026 ms · 2026-05-22T13:45:44.064804+00:00 · methodology

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/RealityFromDistinction.lean reality_from_one_distinction unclear

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

RCgr(C) ... bounded directed-complete ordered set, while GCgr(C) ... algebraic lattice ... Scott continuous morphism ... relative adjunctions between RCgr(C) and GCgr(C) as well as between RCCat and GCCat
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Riguet congruence Φ = (Φob, Φfl) ... conditions (a)–(f) ... quotient category C/Φ

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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.
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