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The Flat Cover Conjecture for Monoid Acts

Sean Cox

TL;DR

This work extends the Flat Cover Conjecture to the non-additive category Act-$S$ by establishing a top-down, set-theoretic criterion for cofibrant generation based on almost-everywhere effectiveness. Under the assumption that $S$ is right-reversible and flat $S$-acts are closed under stable Rees extensions, the FCC holds in Act-$S$, with a stronger conclusion for Act$_0$-$S$ when a left-zero exists. A central tool is a new characterization that cofibrant generation of a class of monomorphisms is equivalent to almost-everywhere effectiveness, enabling streamlined proofs that avoid transfinite constructions. The paper also analyzes strongly flat acts and unitary monomorphisms, showing how cofibrant generation imposes bounds on indecomposable objects and, in the unitary case, is governed by such bounds. Collectively, these results illuminate when FCC and its strong variants hold in monoid-act categories and point to a set-theoretic pathway to broader generalizations and conjectures.

Abstract

We prove that the Flat Cover Conjecture holds for the category of (right) acts over any right-reversible monoid $S$, provided that the flat $S$-acts are closed under stable Rees extensions. The argument shows that the class $\mathcal{F}$-Mono ($S$-act monomorphisms with flat Rees quotient) is cofibrantly generated in such categories, answering a question of Bailey and Renshaw. But cofibrant generation of $\mathcal{SF}$-Mono ($S$-act monomorphisms with \emph{strongly} flat Rees quotient) appears much stronger, since we show it implies that there is a bound on the size of the indecomposable strongly flat acts. Similarly, cofibrant generation of $\mathcal{U}_{\mathcal{F}}$ (unitary monomorphisms with flat complement) implies a bound on the size of indecomposable flat acts. The key tool is a new characterization of cofibrant generation of a class of monomorphisms in terms of ``almost everywhere" effectiveness of the class.

The Flat Cover Conjecture for Monoid Acts

TL;DR

This work extends the Flat Cover Conjecture to the non-additive category Act- by establishing a top-down, set-theoretic criterion for cofibrant generation based on almost-everywhere effectiveness. Under the assumption that is right-reversible and flat -acts are closed under stable Rees extensions, the FCC holds in Act-, with a stronger conclusion for Act- when a left-zero exists. A central tool is a new characterization that cofibrant generation of a class of monomorphisms is equivalent to almost-everywhere effectiveness, enabling streamlined proofs that avoid transfinite constructions. The paper also analyzes strongly flat acts and unitary monomorphisms, showing how cofibrant generation imposes bounds on indecomposable objects and, in the unitary case, is governed by such bounds. Collectively, these results illuminate when FCC and its strong variants hold in monoid-act categories and point to a set-theoretic pathway to broader generalizations and conjectures.

Abstract

We prove that the Flat Cover Conjecture holds for the category of (right) acts over any right-reversible monoid , provided that the flat -acts are closed under stable Rees extensions. The argument shows that the class -Mono (-act monomorphisms with flat Rees quotient) is cofibrantly generated in such categories, answering a question of Bailey and Renshaw. But cofibrant generation of -Mono (-act monomorphisms with \emph{strongly} flat Rees quotient) appears much stronger, since we show it implies that there is a bound on the size of the indecomposable strongly flat acts. Similarly, cofibrant generation of (unitary monomorphisms with flat complement) implies a bound on the size of indecomposable flat acts. The key tool is a new characterization of cofibrant generation of a class of monomorphisms in terms of ``almost everywhere" effectiveness of the class.

Paper Structure

This paper contains 14 sections, 18 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Monoid acts and covers
  3. A new characterization of cellular generation
  4. Cofibrant and cellular generation
  5. Set theory preliminaries
  6. The new characterization
  7. Proof of Theorems \ref{['thm_Fmono_cofibgen']} and \ref{['thm_LO']}
  8. Proof of Theorem \ref{['thm_Fmono_cofibgen']}
  9. Proof of Theorem \ref{['thm_LO']}
  10. Proof of Theorem \ref{['thm_SF']}
  11. Proof of Theorem \ref{['thm_SF']}, \ref{['item_SF_MonoCofibGen']} $\implies$ \ref{['item_SF_LeftCollapseIndec']}
  12. Proof of Theorem \ref{['thm_SF']}, \ref{['item_SF_LeftCollapseIndec']} $\implies$ \ref{['item_SF_MonoCofibGen']}
  13. Proof of Theorem \ref{['thm_UnitaryFlat']}
  14. Some Questions

Key Result

Theorem 1.3

The following are equivalent for any monoid $S$. If $S$ has a left zero (which implies right-reversibility), the following are equivalent:

Theorems & Definitions (33)

  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1: MR3251751, Theorem 4.11
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 23 more