arXiv subjects

Phil Pützstück

Publications and source records attributed to Phil Pützstück.

6 recordsLinked to original sources

arXiv2026

Cocompactness and Presentability

We give a short proof that $κ$-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable $\infty$-categories. A consequence is that an $\infty$-category $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^\mathsf{op}$ are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable $\infty$-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.

math.CT↗

arXiv2026

Tensor Nilpotence and the Size of the Bousfield Lattice

For each prime $p$ and integer $n \geq 1$, we construct a $p$-local spectrum $X_n$ with $X_n^{\otimes n} \not\simeq 0$ but $X_n^{\otimes (n+1)} \simeq 0$. This refutes the retract conjecture of Hovey-Palmieri. Our construction also allows us to determine the cardinality of the Bousfield lattice to be $2^{2^{\aleph_0}}$, answering a question of Dwyer-Palmieri.

math.AT↗

arXiv2026

Unbounded Weight Structures: (Re)construction and Completion

We develop a theory of completeness for weight structures on stable categories, dual to the theory of complete $t$-structures. As in the bounded case, we show that complete weight structures are determined by their weight heart, giving rise to a universal construction $A \mapsto K(A)$ that assigns a complete weight category to an additive category and recovers classical examples such as homotopy categories of chain complexes. We also give a general construction of weight structures on presentable stable categories generated by a small set of objects, generalizing a result of Bondarko and Pauksztello. This recovers the standard weight structure on spectra and an exotic one related to Anderson duality. We identify their completions with modules over the (spectral) integral Steenrod algebra. To treat naturally occurring examples - such as derived categories of abelian categories and module categories over ring spectra - which are often only partially weight complete, we introduce the notion of weak $t$-structures. Within this framework, we prove that any stable category equipped with compatible weight and weak $t$-structures, and satisfying left weight completeness and right $t$-completeness, can be reconstructed from its heart via a two-step completion process $A \mapsto \widehat{K}(A)$.

math.AT↗

arXiv2026

Norms in equivariant homotopy theory

We show that the $\infty$-category of normed algebras in genuine $G$-spectra, as introduced by Bachmann-Hoyois, is modelled by strictly commutative algebras in $G$-symmetric spectra for any finite group $G$. We moreover provide an analogous description of Schwede's ultra-commutative global ring spectra in higher categorical terms. Using these new descriptions, we exhibit the $\infty$-category of ultra-commutative global ring spectra as a partially lax limit of the $\infty$-categories of genuine $G$-spectra for varying $G$, in analogy with the non-multiplicative comparison of Nardin, Pol, and the second author. Along the way, we establish various new results in parametrized higher algebra, which we hope to be of independent interest.

math.AT↗

arXiv2025

Global Picard Spectra and Borel Parametrized Algebra

We answer a question of Schwede on the existence of global Picard spectra associated to his ultra-commutative global ring spectra; given an ultra-commutative global ring spectrum $R$, we show there exists a global spectrum $\mathrm{pic}_\mathrm{eq}(R)$ assembling the Picard spectra of all underlying $G$-equivariant ring spectra $\mathrm{res}_G R$ of $R$ into one object, in that for all finite groups $G$, the genuine fixed points are given by $\mathrm{pic}_\mathrm{eq}(R)^G \simeq \mathrm{pic}(\mathrm{Mod}_{\mathrm{res}_G R}(\mathrm{Sp}_G))$. Along the way, we develop a generalization of Borel-equivariant objects in the setting of parametrized higher algebra. We use this to assemble the symmetric monoidal categories of $G$-spectra for all finite groups $G$ together with all restrictions and norms into a single `normed global category', and build a comparison functor which allows us to import ultra-commutative $G$-equivariant or global ring spectra into the setting of parametrized higher algebra.

math.AT↗

arXiv2021

Logics of Dependence and Independence: The Local Variants

Modern logics of dependence and independence are based on team semantics, which means that formulae are evaluated not on a single assignment of values to variables, but on a set of such assignments, called a team. This leads to high expressive power, on the level of existential second-order logic. As an alternative, Baltag and van Benthem have proposed a local variant of dependence logic, called logic of functional dependence (LFD). While its semantics is also based on a team, the formulae are evaluated locally on just one of its assignments, and the team just serves as the supply of the possible assignments that are taken into account in the evaluation process. This logic thus relies on the modal perspective of generalized assignments semantics, and can be seen as a fragment of first-order logic. For the variant of LFD without equality, the satisfiability problem is decidable. We extend the idea of localising logics of dependence and independence in a systematic way, taking into account local variants of standard atomic dependency properties: besides dependence and independence, also inclusion, exclusion, and anonymity. We study model-theoretic and algorithmic questions of the localised logics, and also resolve some of the questions that had been left open by Baltag and van Benthem. In particular, we study decidability issues of the local logics, and prove that satisfiability of LFD with equality is undecidable. Further, we establish characterisation theorems via appropriate notions of bisimulation and study the complexity of model checking problems for these logics.

cs.LO↗