45 research outputs found

    Logical Berkovich Geometry: A Point-free Perspective

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    Extending our insights from \cite{NVOstrowski}, we apply point-free techniques to sharpen a foundational result in Berkovich geometry. In our language, given the ring A:=K{R−1T}\mathcal{A}:=K\{R^{-1}T\} of convergent power series over a suitable non-Archimedean field KK, the points of its Berkovich Spectrum M(A)\mathcal{M}(\mathcal{A}) correspond to RR-good filters. The surprise is that, unlike the original result by Berkovich, we do not require the field KK to be non-trivially valued. Our investigations into non-Archimedean geometry can be understood as being framed by the question: what is the relationship between topology and logic

    Adelic Geometry via Topos Theory

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    Our starting point has to do with a key tension running through number theory: although all completions of the rationals Q should be treated symmetrically, this is complicated by fundamental disanalogies between the p-adics vs. the reals. Whereas prior work has typically been guided by classical point-set reasoning, this thesis explores various ways of pulling this problem away from the underlying set theory, revealing various surprises that are obscured by the classical perspective. Framing these investigations is the following test problem: construct and describe the topos of completions of Q (up to equivalence). Chapter 2 begins with the preliminaries: we set up the topos-theoretic framework of point-free topology, with a view towards highlighting the distinction between classical vs. geometric mathematics, before introducing the number-theoretic context. A key theme is that geometric mathematics possesses an intrinsic continuity, which forces us to think more carefully about the topological character of classical algebraic constructions. Chapter 3 represents the first step towards constructing the topos of completions. Here, we provide a pointfree account of real exponentiation and logarithms, which will allow us to define the equivalence of completions geometrically. Chapter 4 provides a geometric proof of Ostrowski's Theorem for both upper-valued abosolute values on Z as well as Dedekind-valued absolute values on Q, along with some key insights about the relationship between the multiplicative seminorms and upper reals. In a slightly more classical interlude, Chapter 5 extends these insights to obtain a surprising generalisation of a foundational result in Berkovich geometry. Namely, by replacing the use of classical rigid discs with formal balls, we obtain a classification of the points of Berkovich Spectra M(K{R^{-1}T}) via the language of filters [more precisely, what we call: R-good filters] even when the base field K is trivially-valued. Returning to geometricity, Chapter 6 builds upon Chapters 3 and 4 to investigate the space of places of Q via descent arguments. Here, we uncover an even deeper surprise. Although the non-Archimedean places correspond to singletons (as is classically expected), the Archimedean place corresponds to the subspace of upper reals in [0, 1], a sort of blurred unit interval. The chapter then analyses the topological differences between the non-Archimedean vs. Archimedean places. In particular, we discover that while the topos corresponding to Archimedean place witnesses non-trivial forking in the connected components of its sheaves, the topos corresponding to the non-Archimedean place eliminates all kinds of forking phenomena. We then conclude with some insights and observations, framed by the question: "How should the connected and the disconnected interact?

    Einstein vs. Bergson

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    On 6 April 1922, Einstein met Bergson to debate the nature of time: is the time the physicist calculates the same time the philosopher reflects on? Einstein claimed that only scientific time is real, while Bergson argued that scientific time always presupposes a living and perceiving subject. On that day, nearly 100 years ago, conflict was inevitable. Is it still inevitable today? How many kinds of time are there

    The finite dual coalgebra as a quantization of the maximal spectrum

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    In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras AA as those with enough finite-dimensional representations to let A∘A^\circ act as an appropriate depiction of the noncommutative maximal spectrum of AA; importantly, this class includes affine noetherian PI algebras. We investigate cases where the finite dual coalgebra of a twisted tensor product is a crossed product coalgebra of the respective finite duals. This is achieved by interpreting the finite dual as a topological dual. Sufficient conditions for this result to be applied to Ore extensions, smash product algebras, and crossed product bialgebras are described. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. Finally, we implement these techniques for quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions.Comment: 56 pages, 2 figures. Some corrections made to Subsection 2.2. Comments appreciated

    Exponentiable Grothendieck categories in flat Algebraic Geometry

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    We introduce and describe the 22-category Grt♭\mathsf{Grt}_{\flat} of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories ⊠\boxtimes restricts nicely to Grt♭\mathsf{Grt}_{\flat}. Then, we characterize exponentiable objects with respect to ⊠\boxtimes: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme XX, the category of quasi-coherent sheaves Qcoh(X)\mathsf{Qcoh}(X) is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.Comment: Minor revision. The proofs of Sec 5 have been expanded to make the paper self containe

    Toposes of monoid actions

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    openWe study toposes of actions of monoids on sets. We begin with ordinary actions, producing a class of presheaf toposes which we characterize. As groundwork for considering topological monoids, we branch out into a study of supercompactly generated toposes (a class strictly larger than presheaf toposes). This enables us to efficiently study and characterize toposes of continuous actions of topological monoids on sets, where the latter are viewed as discrete spaces. Finally, we refine this characterization into necessary and sufficient conditions for a supercompactly generated topos to be equivalent to a topos of this form.openInformatica e matematica del calcoloRogers, Morga

    A logical study of some 2-categorical aspects of topos theory

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    There are two well-known topos-theoretic models of point-free generalized spaces: the original Grothendieck toposes (relative to classical sets), and a relativized version (relative to a chosen elementary topos S S with a natural number object) in which the generalized spaces are the bounded geometric morphisms from an elementary topos E E to S S , and they form a 2-category BTop/S BTop/S . However, often it is not clear what a preferred choice for the base S S should be. In this work, we review and further investigate a third model of generalized spaces, based on the 2-category Con Con of ‘contexts for Arithmetic Universes (AUs)’ presented by AU-sketches which originally appeared in Vickers’ work in [Vic19] and [Vic17]. We show how to use the AU techniques to get simple proofs of conceptually stronger, base-independent, and predicative (op)fibration results in ETop ETop , the 2-category of elementary toposes equipped with a natural number object, and arbitrary geometric morphisms. In particular, we relate the strict Chevalley fibrations, used to define fibrations of AU-contexts, to non-strict Johnstone fibrations, used to define fibrations of toposes. Our approach brings to light the close connection of (op)fibration of toposes, conceived as generalized spaces, with topological properties. For example, every local homeomorphism is an opfibration and every entire map (i.e. fibrewise Stone) is a fibration
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