We introduce the Zariski frame of any presentably symmetric monoidal $\infty$-category. This allows us to unify several spectral theories arising in higher algebra. The Zariski frame is coherent whenever the category is compactly generated, and the associated spectral space recovers both the classical Zariski spectrum of a commutative ring and the Hochster dual of the Balmer spectrum of a commutative $2$-ring. Moreover, the smashing frame of any stable presentably symmetric monoidal $\infty$-category can be identified with the Zariski frame of its category of dualizable modules. This construction is based on the principle that ideals in a symmetric monoidal $\infty$-category should be understood as monomorphisms into the unit object. In suitable contexts, this notion recovers the kinds of ideals appearing in the preceding examples, including thick ideals and smashing ideals, and it also accommodates the smashing ideals of non-stable $\infty$-categories. We also study the problem of forming quotients by ideals, which is subtle in the setting of higher algebra. To address this, we introduce two properties of pointed $\infty$-categories, called $Σ$-triviality and $Σ$-exactness. These
We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods.
We define a class of smashing localisations which we call compactly central, and classify compactly central localisations of $Sp_{(p)}$ and of $Sp$. Our main result is that $L_n^f$ is a compactly central localisation. A map $α: 1 \to A$ in a presentably symmetric monoidal $\infty$-category $\mathscr{C}$ is central if there exists a homotopy $α\otimes id_A \simeq id_A \otimes α: A \to A \otimes A$. A central map $α$ can be used to produce a smashing localisation $L_α$ of $\mathscr{C}$, because the free $\mathbb{E}_1$ algebra on the $\mathbb{E}_0$ algebra $α$ is an idempotent commutative algebra. When both the monoidal unit and $A$ are compact, we call $L_α$ compactly central. We show that when $\mathscr{C}$ is (compactly generated) rigid, all compactly central localisations are finite in the sense of Miller. Not all finite localisations of $Sp$ are compactly central. To exhibit $L_n^f$ as compactly central, we determine properties of the $K(n)$-homology of a map between $p$-local finite spectra which ensure that some tensor power of the map is central.
High-energy nuclear collisions have recently emerged as a promising ``imaging-by-smashing'' approach that may reveal the intrinsic shapes of atomic nuclei. Here, I outline a conceptual framework for this technique, explaining how nuclear shapes are encoded during quark-gluon plasma formation and evolution, and how they can be decoded from final-state particle distributions. I highlight the method's potential to advance our understanding of both nuclear structure and quark-gluon plasma physics.
For an arbitrary $\infty$-topos, we classify the smashing localizations in the $\infty$-category of sheaves valued in derived vector spaces: Any of them is the restriction functor to a (unique) closed subtopos. Our proof is based on the existence of a Boolean cover. This result in particular gives us the first example of a nonzero presentably symmetric monoidal stable $\infty$-category whose smashing spectrum has no points. Combining this with the sheaves-spectrum adjunction, we obtain a Tannaka-type categorical reconstruction result for locales.
Building on results of Bazzoni-Šťovíček, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. In particular, we give an explicit construction of an infinite family of commutative rings such that the telescope conjecture fails and which generalise an example of Keller. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the Krull dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories.
WebAssembly is an instruction set architecture and binary format standard, designed for secure execution by an interpreter. Previous work has shown that WebAssembly is vulnerable to buffer overflow due to the lack of effective protection mechanisms. In this paper, we evaluate the implementation of Stack Smashing Protection (SSP) in WebAssembly standalone runtimes, and uncover two weaknesses in their current implementation. The first one is the possibility to overwrite the SSP reference value because of the contiguous memory zones inside a WebAssembly process. The second comes from the reliance of WebAssembly on the runtime to provide randomness in order to initialize the SSP reference value, which impacts the robustness of the solution. We address these two flaws by hardening the SSP implementation in terms of storage and random generator failure, in a way that is generalizable to all of WebAssembly. We evaluate our new, more robust, solution to prove that the implemented improvements do not reduce the efficiency of SSP.
We develop the theory of stratification for a rigidly-compactly generated tensor-triangulated category using the smashing spectrum and the small smashing support. Within the stratified context, we investigate connections between big prime ideals, objectwise-prime ideals and homological primes, and we show that the Telescope Conjecture holds if and only if the homological spectrum is $T_0$ and the homological support detects vanishing. We also reduce stratification to smashing localizations. Moreover, we study induced maps between smashing spectra and prove a descent theorem for stratification. Outside the stratified context, we prove that the Telescope Conjecture holds if and only if the smashing spectrum is $T_0$ with respect to the small topology.
We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$-equivariant Johnson-Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations in the context of a quasi-Galois extension of $E_\infty$-rings.
We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it holds precisely for those Prüfer domains which are strongly discrete.
Given a tensor-triangulated category $T$, we prove that every flat tensor-idempotent in the module category over $T^c$ (the compacts) comes from a unique smashing ideal in $T$. We deduce that the lattice of smashing ideals forms a frame.
We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman. As another consequence, we obtain a cofinite type result for pure-injective cosilting objects. We also give a formulation of telescope conjecture for homotopically smashing t-structures in underlying triangulated categories of certain Grothendieck derivators.
We study smashing subcategories of a triangulated category with coproducts via silting theory. Our main result states that for derived categories of dg modules over a non-positive differential graded ring, every compactly generated localising subcategory is generated by a partial silting object. In particular, every such smashing subcategory admits a silting t-structure.
Recognizing subtle historical patterns is central to modeling and forecasting problems in time series analysis. Here we introduce and develop a new approach to quantify deviations in the underlying hidden generators of observed data streams, resulting in a new efficiently computable universal metric for time series. The proposed metric is in the sense that we can compare and contrast data streams regardless of where and how they are generated and without any feature engineering step. The approach proposed in this paper is conceptually distinct from our previous work on data smashing, and vastly improves discrimination performance and computing speed. The core idea here is the generalization of the notion of KL divergence often used to compare probability distributions to a notion of divergence in time series. We call this the sequence likelihood (SL) divergence, which may be used to measure deviations within a well-defined class of discrete-valued stochastic processes. We devise efficient estimators of SL divergence from finite sample paths and subsequently formulate a universal metric useful for computing distance between time series produced by hidden stochastic generators.
The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K-theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form.
Researchers in the Gulf of California have documented a dramatic orca behavior never described before。 In two separate encounters, one orca gripped a dead sunfish by the fin while another charged into it at high speed, blasting the carcass into thousands of pieces
Investigation of the underlying physics or biology from empirical data requires a quantifiable notion of similarity - when do two observed data sets indicate nearly identical generating processes, and when they do not. The discriminating characteristics to look for in data is often determined by heuristics designed by experts, $e.g.$, distinct shapes of "folded" lightcurves may be used as "features" to classify variable stars, while determination of pathological brain states might require a Fourier analysis of brainwave activity. Finding good features is non-trivial. Here, we propose a universal solution to this problem: we delineate a principle for quantifying similarity between sources of arbitrary data streams, without a priori knowledge, features or training. We uncover an algebraic structure on a space of symbolic models for quantized data, and show that such stochastic generators may be added and uniquely inverted; and that a model and its inverse always sum to the generator of flat white noise. Therefore, every data stream has an anti-stream: data generated by the inverse model. Similarity between two streams, then, is the degree to which one, when summed to the other's anti
In Homotopy Type Theory, few constructions have proved as troublesome as the smash product. While its definition is just as direct as in classical mathematics, one quickly realises that in order to define and reason about functions over iterations of it, one has to verify an exponentially growing number of coherences. This has led to crucial results concerning smash products remaining open. One particularly important such result is the fact that smash products form a (1-coherent) symmetric monoidal product on the universe of pointed types. This fact was used, without a complete proof, by e.g. Brunerie in his PhD thesis to construct the cup product on integral cohomology and is, more generally, a fundamental result in traditional algebraic topology. In this paper, we salvage the situation by introducing a simple informal heuristic for reasoning about functions defined over iterated smash products. We then use the heuristic to verify e.g. the hexagon and pentagon identities, thereby obtaining a proof of symmetric monoidality. We also provide a formal statement of the heuristic in terms of an induction principle concerning the construction of homotopies of functions defined over itera
SM*A*S*H is an extension of the Standard Model of particle physics which has just the minimal number of fields in order to solve six puzzles of particle physics and cosmology in one smash: vacuum stability, inflation, baryon asymmetry, neutrino masses, strong CP, and dark matter. The parameters of SM*A*S*H are constrained by symmetries and requirements to solve these puzzles. This provides various firm predictions for observables which can be confronted with experiments. We discuss the prospects and timeline to scrutinise or smash SM*A*S*H by cosmic microwave background polarisation experiments, axion haloscopes, and future space-borne gravitational wave detectors.
We discuss cases where Voevodsky's smash nilpotence conjecture is known, and give a few new ones. In particular we explain a theorem of the cube for $1$-cycles, which is due to Oussama Ouriachi.