4,573 research outputs found

    Kan extensions in probability theory

    Get PDF
    n this thesis we will discuss results and ideas in probability theory from a categorical point of view. One categorical concept in particular will be of interest to us, namely that of Kan extensions. We will use Kan extensions of ‘ordinary’ functors, enriched functors and lax natural transformations to give categorical proofs of some fundamental results in probability theory and measure theory. We use Kan extensions of ‘ordinary’ functors to represent probability monads as codensity monads. We consider a functor representing probability measures on countable spaces. By Kan extending this functor along itself, we obtain a codensity monad describing probability measures on all spaces. In this way we represent probability monads such as the Giry monad, the Radon monad and the Kantorovich monad. Kan extensions of lax natural transformations are used to obtain a categorical proof of the Carath´eodorody extensions theorem. The Carath´eodory extension theorem is a fundamental theorem in measure theory that says that premeasures can be extended to measures. We first develop a framework for Kan extensions of lax natural transformations. We then represent outer and inner (pre)measures by certain lax and colax natural transformations. By applying the results on extensions of transformations a categorical proof of Carath´eodory’s extension theorem is obtained. We also give a categorical view on the Radon–Nikodym theorem and martingales. For this we need Kan extensions of enriched functors. We start by observing that the finite version of the Radon–Nikodym theorem is trivial and that it can be interpreted as a natural isomorphism between certain functors, enriched over CMet, the category of complete metric spaces and 1-Lipschitz maps. We proceed by Kan extending these, to obtain the general version of the Radon–Nikodym theorem. Concepts such as conditional expectation and martingales naturally appear in this construction. By proving that these extended functors preserve certain cofiltered limits, we obtain categorical proofs of a weaker version of a martingale convergence theorem and the Kolmogorov extension theorem

    C*-Correspondences, Hilbert Bimodules, and their L^p Versions

    Get PDF
    This dissertation initiates the study of LpL^p-modules, which are modules over LpL^p-operator algebras inspired by Hilbert modules over C*-algebras. The primary motivation for studying LpL^p-modules is to explore the possibility of defining LpL^p analogues of Cuntz-Pimsner algebras. The first part of this thesis consists of investigating representations of C*-correspondences on pairs of Hilbert spaces. This generalizes the concept of representations of Hilbert bimodules introduced by R. Exel in \cite{Exel1993}. We present applications of representing a correspondence on a pair of Hilbert spaces (\Hi_0, \Hi_1), such as obtaining induced representations of both \Li_A(\X) and \mathcal{K}_A(\X) on \Hi_1, and giving necessary and sufficient conditions on an (A,B)(A,B) C*-correspondences to admit a Hilbert AA-BB-bimodule structure. The second part is concerned with the theory of LpL^p-modules. Here we present a thorough treatment of LpL^p-modules, including morphisms between them and techniques for constructing new LpL^p-modules. We then useour results on representations for C*-correspondences to motivate and develop the theory of LpL^p-correspondences, their representations, the LpL^p-operator algebras they generate, and present evidence that well-known LpL^p-operator algebras can be constructed from LpL^p-correspondences via LpL^p-Fock representations. Due to the technicality that comes with dealing with direct sums of LpL^p-correspondences and interior tensor products, we only focus on two particular examples for which a Fock space construction can be carried out. The first example deals with the LpL^p-module (ℓdp,ℓdq)(\ell_d^p, \ell_d^q), for which we exhibit a covariant LpL^p-Fock representation that yields an LpL^p-operator algebra isometrically isomorphic to Odp\mathcal{O}_d^p, the LpL^p-analogue of the Cuntz-algebra Od\mathcal{O}_d introduced by N.C. Phillips in \cite{ncp2012AC}. The second example involves a nondegenerate LpL^p-operator algebra AA with a bicontractive approximate identity together with an isometric automorphism \varphi_A \in \op{Aut}(A). In this case, we also present an algebra associated to a covariant LpL^p-Fock representation, but due to the current lack of knowledge of universality of the LpL^p-Fock representation, we only show that there is a contractive map from the crossed product Fp(Z,A,φA)F^p(\Z, A, \varphi_A) to this algebra. This dissertation includes unpublished material

    A functorial formalism for quasi-coherent sheaves on a geometric stack

    Get PDF
    A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks, an adjunction f^∗⊣f_∗ for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid.Ministerio de Ciencia e Innovación | Ref. MTM2008-03465Ministerio de Ciencia e Innovación | Ref. MTM2011-26088Xunta de Galicia | Ref. GRC2013-04

    The Ontology of Haag’s Local Quantum Physics

    Get PDF
    The ontology of Local Quantum Physics, Rudolf Haag’s framework for relativistic quantum theory, is reviewed and discussed. It is one of spatiotemporally localized events and unlocalized causal intermediaries, including the elementary particles, which come progressively into existence in accordance with a fundamental arrow of time. Haag’s conception of quantum theory is distinguished from others in which events are also central, especially those of Niels Bohr and John Wheeler, with which it has been compared

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Slopes of modular forms and geometry of eigencurves

    Full text link
    Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's ⌊k−1p+1⌋\lfloor\frac{k-1}{p+1}\rfloor-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture.Comment: 97 pages; comments are welcom

    Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology

    Full text link
    We study smooth higher symmetry groups and moduli ∞\infty-stacks of generic higher geometric structures on manifolds. Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold. We construct the smooth higher symmetry group of any geometric structure on MM and show that this completely classifies, via a universal property, equivariant structures on the higher geometry. We construct moduli stacks of higher geometric data as ∞\infty-categorical quotients by the action of the higher symmetries, extract information about the homotopy types of these moduli ∞\infty-stacks, and prove a helpful sufficient criterion for when two such higher moduli stacks are equivalent. In the second part of the paper we study higher U(1)\mathrm{U}(1)-connections. First, we observe that higher connections come organised into higher groupoids, which further carry affine actions by Baez-Crans-type higher vector spaces. We compute a presentation of the higher gauge actions for nn-gerbes with kk-connection, comment on the relation to higher-form symmetries, and present a new String group model. We construct smooth moduli ∞\infty-stacks of higher Maxwell and Einstein-Maxwell solutions, correcting previous such considerations in the literature, and compute the homotopy groups of several moduli ∞\infty-stacks of higher U(1)\mathrm{U}(1)- connections. Finally, we show that a discrepancy between two approaches to the differential geometry of NSNS supergravity (via generalised and higher geometry, respectively) vanishes at the level of moduli ∞\infty-stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom

    On Vector Spaces with Formal Infinite Sums

    Full text link
    We discuss possible definitions of categories of vector spaces enriched with a notion of formal infinite linear combination in the likes of the formal infinite linear combinations one has in the context of generalized power series, we call these \textit{categories of reasonable strong vector spaces} (r.s.v.s.). We show that, in a precise sense, the more general possible definition for a strong vector space is that of a small Vect\mathrm{Vect}-enriched endofunctor of Vect\mathrm{Vect} that is right orthogonal, for every cardinal λ\lambda, to the cokernel of the canonical inclusion of the λ\lambda-th copower in the λ\lambda-th power of the identity functor: these form the objects for a universal r.s.v.s. we call ΣVect\Sigma\mathrm{Vect}. We relate this category to what could be understood to be the obvious category of strong vector spaces BΣVectB\Sigma\mathrm{Vect} and to the r.s.v.s. KTVectsK\mathrm{TVect}_s of separated linearly topologized spaces that are generated by linearly compact spaces. We study the number of iterations of the obvious approximate reflector on Ind-(Vectop)\mathrm{Ind}\text{-}(\mathrm{Vect}^{\mathrm{op}}) needed to construct the orthogonal reflector Ind-(Vectop)→ΣVect\mathrm{Ind}\text{-}(\mathrm{Vect}^{\mathrm{op}}) \to \Sigma\mathrm{Vect} as it relates to the problem of constructing the smallest subspace of an X∈ΣVectX \in \Sigma\mathrm{Vect} closed under taking infinite linear combinations containing a given linear subspace of HH of XX. Finally we show the natural monoidal closed structure on Ind-(Vectop)\mathrm{Ind}\text{-}(\mathrm{Vect}^{\mathrm{op}}) restricts naturally to ΣVect\Sigma\mathrm{Vect} and apply this to define an infinite-sum-sensitive notion of K\"ahler differentials for generalized power series. Most of the technical results apply to a more general class of orthogonal subcategories of Ind-(Vectop)\mathrm{Ind}\text{-}(\mathrm{Vect}^{\mathrm{op}}) and we work with that generality

    Deformation theory of G-valued pseudocharacters and symplectic determinant laws

    Get PDF
    We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and Lafforgue. We refer to Taylor’s and Rouquier’s pseudorepresentations as pseudocharacters. They are very closely related, the main difference being that Taylor’s pseudocharacters are defined for a group, where as Rouquier’s pseudocharacters are defined for algebras. Chenevier’s pseudorepresentations are so-called polynomial laws and will be called determinant laws. Lafforgue’s pseudorepresentations are a generalization of Taylor’s pseudocharacters to other reductive groups G, in that the corresponding notion of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters. We survey the known comparison theorems, notably Emerson’s bijection between Chenevier’s determinant laws and Lafforgue’s GL(n)-pseudocharacters and the bijection with Taylor’s pseudocharacters away from small characteristics. We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously, we obtain that tensor products of determinant laws exist and are compatible with tensor products of representations. Further the tensor product of Lafforgue’s pseudocharacters agrees with the tensor product of Taylor’s pseudocharacters. We generalize some of the results of [Che14] to general reductive groups, in particular we show that the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely generated profinite group Γ with values in a finite field (of characteristic p) is noetherian. We also show, that for specific groups G it is sufficient, that Γ satisfies Mazur’s condition Φ_p. One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater of Gal(F) into obstructed subloci Xdec(Θ), Xpair(Θ), Xspcl(Θ) of dimension smaller than the expected dimension n(2n + 1)[F : Qp]. We also prove that Lafforgue’s G-pseudocharacters over algebraically closed fields for possibly nonconnected reductive groups G come from a semisimple representation. We introduce a formal scheme and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to conjugacy and Frobenius automorphisms. At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic groups, which is based on adding a ’Pfaffian polynomial law’ to a determinant law which is invariant under an involution. We prove the expected basic properties in that we show that symplectic determinant laws over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgue’s pseudocharacters and show that it is an isomorphism over reduced rings

    Enumerative invariants in self-dual categories. I. Motivic invariants

    Full text link
    In this series of papers, we propose a theory of enumerative invariants counting self-dual objects in self-dual categories. Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group GL(n)\mathrm{GL} (n), and our theory is an extension of this to structure groups O(n)\mathrm{O} (n) and Sp(2n)\mathrm{Sp} (2n). Examples of our invariants include invariants counting principal orthogonal or symplectic bundles, and invariants counting self-dual quiver representations. In the present paper, we take the motivic approach, and define our invariants as elements in a ring of motives. One can also extract numerical invariants from these invariants. We prove wall-crossing formulae relating our invariants for different stability conditions. We also provide an explicit algorithm computing invariants for quiver representations, and we present some numerical results.Comment: 122 page
    • …
    corecore