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Sean Cox

Publications and source records attributed to Sean Cox.

At least 19 recordsLinked to original sources
arXiv2026

Kaplansky classes and stability

Soon after the proof of the Flat Cover Conjecture around the year 2000, two related concepts were introduced for classes in Grothendieck categories: \emph{Deconstructible classes} and the strictly weaker \emph{Kaplansky classes}. All commonly-studied Kaplansky classes, such as the class $\mathcal{FM}$ of Flat Mittag-Leffler modules in $R$-Mod and the class $\mathcal{D}$ of Drinfeld vector bundles in Qcoh($X$), satisfy a stronger property we introduce here: they are \emph{Uniformly Stationary Kaplansky} classes. While such classes generally lack the key feature (existence of precovers) that make deconstructible classes so central to modern relative homological algebra, they often suffice for model-theoretic stability. This is true even in the absence of the Amalgamation Property, with various restricted classes of morphisms, and in some non-additive settings. For example, $\mathcal{FM}$ with pure embeddings, and $\mathcal{D}$ with (either categorical or geometric) pure embeddings, are stable in all sufficiently closed cardinals.

math.LO↗
arXiv2026

Cellular generation revisited

Cellular generation, which generalises cofibrant generation, is an important categorical smallness condition on a class of morphisms. A general challenge is to determine whether a given class of morphisms $\mathcal{M}$ is cellularly generated, in which $\mathcal{M}$-effective squares are often useful. These are commuting squares consisting of morphisms in $\mathcal{M}$, so that the induced morphism from the pushout square is also in $\mathcal{M}$. When we drop the requirement that the vertical morphisms in the square are in $\mathcal{M}$ we obtain the weaker notion of $\mathcal{M}$-quasieffective square. We prove that, in a locally presentable category, $\mathcal{M}$ is cellularly generated if and only if $\mathcal{M}$ is almost everywhere quasieffective. The latter is a set-theoretic condition stating that for almost every partial elementary set-theoretic subuniverse $\mathfrak{N}$, we have that restricting any morphism in $\mathcal{M}$ to $\mathfrak{N}$ yields an $\mathcal{M}$-quasieffective square. For locally finitely presentable categories this yields an additional categorical characterisation in terms of filtrations of $\mathcal{M}$-quasieffective squares. If we additionally assume that $\mathcal{M}$ is continuous (i.e., the corresponding wide subcategory is closed under directed colimits) then we obtain a stronger characterisation of cellular generation in terms of accessibility of the category of $\mathcal{M}$-effective squares. This improves on a theorem by Lieberman, Vasey, and the third author.

math.CT↗
arXiv2025

The Flat Cover Conjecture for Monoid Acts

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.

math.CT↗
arXiv2025

Cofibrant generation of pure monomorphisms in presheaf categories

We characterise when the pure monomorphisms in a presheaf category $\mathbf{Set}^\mathcal{C}$ are cofibrantly generated in terms of the category $\mathcal{C}$. In particular, when $\mathcal{C}$ is a monoid $S$ this characterises cofibrant generation of pure monomorphisms between sets with an $S$-action in terms of $S$: this happens if and only if for all $a, b \in S$ there is $c \in S$ such that $a = cb$ or $ca = b$. We give a model-theoretic proof: we prove that our characterisation is equivalent to having a stable independence relation, which in turn is equivalent to cofibrant generation. As a corollary, we show that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated.

math.CT↗
arXiv2024

Vop\v{e}nka's Principle, Maximum Deconstructibility, and singly-generated torsion classes

Deconstructibility is an often-used sufficient condition on a class $\mathcal{C}$ of modules that allows one to carry out homological algebra \emph{relative to $\mathcal{C}$}. The principle \textbf{Maximum Deconstructibility (MD)} asserts that a certain necessary condition for a class to be deconstructible is also sufficient. MD implies, for example, that the classes of Gorenstein Projective modules, Ding Projective modules, their relativized variants, and all torsion classes are deconstructible over any ring. MD was known to follow from Vop\v{e}nka's Principle and imply the existence of an $\omega_1$-strongly compact cardinal. We prove that MD is equivalent to Vop\v{e}nka's Principle, and to the assertion that each torsion class of abelian groups is generated by a single group within the class (yielding the converse of a theorem of G\"obel and Shelah).

math.LO↗
arXiv2024

Approximation properties of torsion classes

We strengthen a result of Bagaria and Magidor~\cite{MR3152715} about the relationship between large cardinals and torsion classes of abelian groups, and prove that (1) the \emph{Maximum Deconstructibility} principle introduced in \cite{Cox_MaxDecon} requires large cardinals; it sits, implication-wise, between Vop\v{e}nka's Principle and the existence of an $\omega_1$-strongly compact cardinal. (2) While deconstructibility of a class of modules always implies the precovering property by \cite{MR2822215}, the concepts are (consistently) non-equivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.

math.LO↗
arXiv2024

Approximation Theory and Elementary Submodels

\emph{Approximation Theory} uses nicely-behaved subcategories to understand entire categories, just as projective modules are used to approximate arbitrary modules in classical homological algebra. We use set-theoretic \emph{elementary submodel arguments} to give new, short proofs of well-known theorems in approximation theory, sometimes with stronger results.

math.LO↗
arXiv2023

Sparse systems of functions and quasi-analytic classes

We provide a new characterization of quasi-analyticity of Denjoy-Carleman classes, related to \emph{Wetzel's Problem}. We also completely resolve which Denjoy-Carleman classes carry \emph{sparse systems}: if the Continuum Hypothesis (CH) holds, \textbf{all} Denjoy-Carleman classes carry sparse systems; but if CH fails, a Denjoy-Carleman class carries a sparse system if and only if it is not quasi-analytic. As corollaries, we extend results of \cite{MR3552748} and \cite{CodyCoxLee} about non-existence of "anonymous predictors" for real functions.

math.LO↗
arXiv2022

Sparse analytic systems

Erd\H{o}s \cite{MR168482} proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal{F}$ of (real or complex) analytic functions, such that $\big\{ f(x) \ : \ f \in \mathcal{F} \big\}$ is countable for every $x$. We strengthen Erd\H{o}s' result by proving that CH is equivalent to the existence of what we call \emph{sparse analytic systems} of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim$ on $\mathbb{R}$ such that any "analytic-anonymous" attempt to predict the map $x \mapsto [x]_\sim$ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman \cite{MR3552748}.

math.LO↗
arXiv2022

How robustly can you predict the future?

Hardin and Taylor \cite{MR2384262} proved that any function on the reals -- even a nowhere continuous one -- can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed in \cite{MR3100500} that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman \cite{MR3552748}, who provided upper and lower frontiers (in the subgroup lattice of $\text{Homeo}^+(\mathbb{R})$) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of H\"older's Theorem (that every Archimedean group is abelian).

math.LO↗
arXiv2021

The Diagonal Strong Reflection Principle and its fragments

A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated to arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle are analyzed, and consequences are presented. Some of these consequences are ``exact'' versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

math.LO↗
arXiv2021

Salce's problem on cotorsion pairs is undecidable

Salce \cite{MR565595} introduced the notion of a \emph{cotorsion pair} of classes of abelian groups, and asked whether every such pair is \emph{complete} (i.e., has enough injectives and projectives). We prove that it is consistent, relative to the consistency of Vop\v{e}nka's Principle (VP), that the answer is affirmative. Combined with a previous result of Eklof-Shelah \cite{MR2031314}, this shows that Salce's Problem is independent of the ZFC axioms (modulo the consistency of VP).

math.LO↗
arXiv2020

Maximum deconstructibility in module categories

We prove that Vop\v{e}nka's Principle implies that for every class $\mathfrak{X}$ of modules over any ring, the class of \textbf{$\boldsymbol{\mathfrak{X}}$-Gorenstein Projective modules} (\textbf{$\boldsymbol{\mathfrak{X}}$-$\boldsymbol{\mathcal{GP}}$}) is a special precovering class. In particular, it is not possible to prove (unless Vop\v{e}nka's Principle is inconsistent) that there is a ring over which the \textbf{Ding Projectives} ($\boldsymbol{\mathcal{DP}}$) or the \textbf{Gorenstein Projectives} ($\boldsymbol{\mathcal{GP}}$) do not form a precovering class (\v{S}aroch previously obtained this result for the class $\mathcal{GP}$, using different methods). The key innovation is a new "top-down" characterization of \emph{deconstructibility}, which is a well-known sufficient condition for a class to be precovering. We also prove that Vop\v{e}nka's Principle implies, in some sense, the maximum possible amount of deconstructibility in module categories.

math.RT↗
arXiv2020

Forcing axioms and the complexity of non-stationary ideals

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $\omega_2$ to sets of ordinals of countable cofinality is $\Delta_1$-definable by formulas with parameters in $H(\omega_3)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $\omega_2$ and strong forcing axioms that are compatible with CH. Finally, we answer a question of S. Friedman, Wu and Zdomskyyshow by showing that the $\Delta_1$-definability of the non-stationary ideal on $\omega_2$ is compatible with arbitrary large values of the continuum function at $\omega_2$.

math.LO↗
arXiv2020

Compactness versus hugeness at successor cardinals

If $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $\omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $\omega_2$ such that $\mathcal{P}(\omega_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.

math.LO↗
arXiv2020

The $\Pi^1_1 \! \! \downarrow$ L\"owenheim-Skolem-Tarski property of Stationary Logic

Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that the L\"owenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets ($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that the restriction of the LST property to (downward) reflection of $\Pi^1_1$ formulas, which we call the $\Pi^1_1 \! \! \downarrow$-LST property, is equivalent to the \emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with results from \cite{Cox_RP_IS}, this shows that the $\Pi^1_1 \! \! \downarrow$-LST Property for Stationary Logic is strictly weaker than the full LST Property for Stationary Logic, though if CH holds they are equivalent.

math.LO↗
arXiv2019

The $\kappa$-Strongly Proper Forcing Axiom

We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly proper forcing notions which are also $\kappa$-lattice, and then eliminate the need for large cardinals. The proof goes through a natural reflection property for $\kappa$-strongly proper forcings. We also produce a model of this forcing axiom with $2^\kappa$ arbitrarily large, and prove the inconsistency of certain natural strengthenings of the axiom.

math.LO↗